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--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/determinant.py
@@ -0,0 +1,898 @@
+from types import FunctionType
+
+from sympy.core.numbers import Float, Integer
+from sympy.core.singleton import S
+from sympy.core.symbol import uniquely_named_symbol
+from sympy.core.mul import Mul
+from sympy.polys import PurePoly, cancel
+from sympy.functions.combinatorial.numbers import nC
+from sympy.polys.matrices.domainmatrix import DomainMatrix
+
+from .common import NonSquareMatrixError
+from .utilities import (
+    _get_intermediate_simp, _get_intermediate_simp_bool,
+    _iszero, _is_zero_after_expand_mul, _dotprodsimp, _simplify)
+
+
+def _find_reasonable_pivot(col, iszerofunc=_iszero, simpfunc=_simplify):
+    """ Find the lowest index of an item in ``col`` that is
+    suitable for a pivot.  If ``col`` consists only of
+    Floats, the pivot with the largest norm is returned.
+    Otherwise, the first element where ``iszerofunc`` returns
+    False is used.  If ``iszerofunc`` does not return false,
+    items are simplified and retested until a suitable
+    pivot is found.
+
+    Returns a 4-tuple
+        (pivot_offset, pivot_val, assumed_nonzero, newly_determined)
+    where pivot_offset is the index of the pivot, pivot_val is
+    the (possibly simplified) value of the pivot, assumed_nonzero
+    is True if an assumption that the pivot was non-zero
+    was made without being proved, and newly_determined are
+    elements that were simplified during the process of pivot
+    finding."""
+
+    newly_determined = []
+    col = list(col)
+    # a column that contains a mix of floats and integers
+    # but at least one float is considered a numerical
+    # column, and so we do partial pivoting
+    if all(isinstance(x, (Float, Integer)) for x in col) and any(
+            isinstance(x, Float) for x in col):
+        col_abs = [abs(x) for x in col]
+        max_value = max(col_abs)
+        if iszerofunc(max_value):
+            # just because iszerofunc returned True, doesn't
+            # mean the value is numerically zero.  Make sure
+            # to replace all entries with numerical zeros
+            if max_value != 0:
+                newly_determined = [(i, 0) for i, x in enumerate(col) if x != 0]
+            return (None, None, False, newly_determined)
+        index = col_abs.index(max_value)
+        return (index, col[index], False, newly_determined)
+
+    # PASS 1 (iszerofunc directly)
+    possible_zeros = []
+    for i, x in enumerate(col):
+        is_zero = iszerofunc(x)
+        # is someone wrote a custom iszerofunc, it may return
+        # BooleanFalse or BooleanTrue instead of True or False,
+        # so use == for comparison instead of `is`
+        if is_zero == False:
+            # we found something that is definitely not zero
+            return (i, x, False, newly_determined)
+        possible_zeros.append(is_zero)
+
+    # by this point, we've found no certain non-zeros
+    if all(possible_zeros):
+        # if everything is definitely zero, we have
+        # no pivot
+        return (None, None, False, newly_determined)
+
+    # PASS 2 (iszerofunc after simplify)
+    # we haven't found any for-sure non-zeros, so
+    # go through the elements iszerofunc couldn't
+    # make a determination about and opportunistically
+    # simplify to see if we find something
+    for i, x in enumerate(col):
+        if possible_zeros[i] is not None:
+            continue
+        simped = simpfunc(x)
+        is_zero = iszerofunc(simped)
+        if is_zero in (True, False):
+            newly_determined.append((i, simped))
+        if is_zero == False:
+            return (i, simped, False, newly_determined)
+        possible_zeros[i] = is_zero
+
+    # after simplifying, some things that were recognized
+    # as zeros might be zeros
+    if all(possible_zeros):
+        # if everything is definitely zero, we have
+        # no pivot
+        return (None, None, False, newly_determined)
+
+    # PASS 3 (.equals(0))
+    # some expressions fail to simplify to zero, but
+    # ``.equals(0)`` evaluates to True.  As a last-ditch
+    # attempt, apply ``.equals`` to these expressions
+    for i, x in enumerate(col):
+        if possible_zeros[i] is not None:
+            continue
+        if x.equals(S.Zero):
+            # ``.iszero`` may return False with
+            # an implicit assumption (e.g., ``x.equals(0)``
+            # when ``x`` is a symbol), so only treat it
+            # as proved when ``.equals(0)`` returns True
+            possible_zeros[i] = True
+            newly_determined.append((i, S.Zero))
+
+    if all(possible_zeros):
+        return (None, None, False, newly_determined)
+
+    # at this point there is nothing that could definitely
+    # be a pivot.  To maintain compatibility with existing
+    # behavior, we'll assume that an illdetermined thing is
+    # non-zero.  We should probably raise a warning in this case
+    i = possible_zeros.index(None)
+    return (i, col[i], True, newly_determined)
+
+
+def _find_reasonable_pivot_naive(col, iszerofunc=_iszero, simpfunc=None):
+    """
+    Helper that computes the pivot value and location from a
+    sequence of contiguous matrix column elements. As a side effect
+    of the pivot search, this function may simplify some of the elements
+    of the input column. A list of these simplified entries and their
+    indices are also returned.
+    This function mimics the behavior of _find_reasonable_pivot(),
+    but does less work trying to determine if an indeterminate candidate
+    pivot simplifies to zero. This more naive approach can be much faster,
+    with the trade-off that it may erroneously return a pivot that is zero.
+
+    ``col`` is a sequence of contiguous column entries to be searched for
+    a suitable pivot.
+    ``iszerofunc`` is a callable that returns a Boolean that indicates
+    if its input is zero, or None if no such determination can be made.
+    ``simpfunc`` is a callable that simplifies its input. It must return
+    its input if it does not simplify its input. Passing in
+    ``simpfunc=None`` indicates that the pivot search should not attempt
+    to simplify any candidate pivots.
+
+    Returns a 4-tuple:
+    (pivot_offset, pivot_val, assumed_nonzero, newly_determined)
+    ``pivot_offset`` is the sequence index of the pivot.
+    ``pivot_val`` is the value of the pivot.
+    pivot_val and col[pivot_index] are equivalent, but will be different
+    when col[pivot_index] was simplified during the pivot search.
+    ``assumed_nonzero`` is a boolean indicating if the pivot cannot be
+    guaranteed to be zero. If assumed_nonzero is true, then the pivot
+    may or may not be non-zero. If assumed_nonzero is false, then
+    the pivot is non-zero.
+    ``newly_determined`` is a list of index-value pairs of pivot candidates
+    that were simplified during the pivot search.
+    """
+
+    # indeterminates holds the index-value pairs of each pivot candidate
+    # that is neither zero or non-zero, as determined by iszerofunc().
+    # If iszerofunc() indicates that a candidate pivot is guaranteed
+    # non-zero, or that every candidate pivot is zero then the contents
+    # of indeterminates are unused.
+    # Otherwise, the only viable candidate pivots are symbolic.
+    # In this case, indeterminates will have at least one entry,
+    # and all but the first entry are ignored when simpfunc is None.
+    indeterminates = []
+    for i, col_val in enumerate(col):
+        col_val_is_zero = iszerofunc(col_val)
+        if col_val_is_zero == False:
+            # This pivot candidate is non-zero.
+            return i, col_val, False, []
+        elif col_val_is_zero is None:
+            # The candidate pivot's comparison with zero
+            # is indeterminate.
+            indeterminates.append((i, col_val))
+
+    if len(indeterminates) == 0:
+        # All candidate pivots are guaranteed to be zero, i.e. there is
+        # no pivot.
+        return None, None, False, []
+
+    if simpfunc is None:
+        # Caller did not pass in a simplification function that might
+        # determine if an indeterminate pivot candidate is guaranteed
+        # to be nonzero, so assume the first indeterminate candidate
+        # is non-zero.
+        return indeterminates[0][0], indeterminates[0][1], True, []
+
+    # newly_determined holds index-value pairs of candidate pivots
+    # that were simplified during the search for a non-zero pivot.
+    newly_determined = []
+    for i, col_val in indeterminates:
+        tmp_col_val = simpfunc(col_val)
+        if id(col_val) != id(tmp_col_val):
+            # simpfunc() simplified this candidate pivot.
+            newly_determined.append((i, tmp_col_val))
+            if iszerofunc(tmp_col_val) == False:
+                # Candidate pivot simplified to a guaranteed non-zero value.
+                return i, tmp_col_val, False, newly_determined
+
+    return indeterminates[0][0], indeterminates[0][1], True, newly_determined
+
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _berkowitz_toeplitz_matrix(M):
+    """Return (A,T) where T the Toeplitz matrix used in the Berkowitz algorithm
+    corresponding to ``M`` and A is the first principal submatrix.
+    """
+
+    # the 0 x 0 case is trivial
+    if M.rows == 0 and M.cols == 0:
+        return M._new(1,1, [M.one])
+
+    #
+    # Partition M = [ a_11  R ]
+    #                  [ C     A ]
+    #
+
+    a, R = M[0,0],   M[0, 1:]
+    C, A = M[1:, 0], M[1:,1:]
+
+    #
+    # The Toeplitz matrix looks like
+    #
+    #  [ 1                                     ]
+    #  [ -a         1                          ]
+    #  [ -RC       -a        1                 ]
+    #  [ -RAC     -RC       -a       1         ]
+    #  [ -RA**2C -RAC      -RC      -a       1 ]
+    #  etc.
+
+    # Compute the diagonal entries.
+    # Because multiplying matrix times vector is so much
+    # more efficient than matrix times matrix, recursively
+    # compute -R * A**n * C.
+    diags = [C]
+    for i in range(M.rows - 2):
+        diags.append(A.multiply(diags[i], dotprodsimp=None))
+    diags = [(-R).multiply(d, dotprodsimp=None)[0, 0] for d in diags]
+    diags = [M.one, -a] + diags
+
+    def entry(i,j):
+        if j > i:
+            return M.zero
+        return diags[i - j]
+
+    toeplitz = M._new(M.cols + 1, M.rows, entry)
+    return (A, toeplitz)
+
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _berkowitz_vector(M):
+    """ Run the Berkowitz algorithm and return a vector whose entries
+        are the coefficients of the characteristic polynomial of ``M``.
+
+        Given N x N matrix, efficiently compute
+        coefficients of characteristic polynomials of ``M``
+        without division in the ground domain.
+
+        This method is particularly useful for computing determinant,
+        principal minors and characteristic polynomial when ``M``
+        has complicated coefficients e.g. polynomials. Semi-direct
+        usage of this algorithm is also important in computing
+        efficiently sub-resultant PRS.
+
+        Assuming that M is a square matrix of dimension N x N and
+        I is N x N identity matrix, then the Berkowitz vector is
+        an N x 1 vector whose entries are coefficients of the
+        polynomial
+
+                        charpoly(M) = det(t*I - M)
+
+        As a consequence, all polynomials generated by Berkowitz
+        algorithm are monic.
+
+        For more information on the implemented algorithm refer to:
+
+        [1] S.J. Berkowitz, On computing the determinant in small
+            parallel time using a small number of processors, ACM,
+            Information Processing Letters 18, 1984, pp. 147-150
+
+        [2] M. Keber, Division-Free computation of sub-resultants
+            using Bezout matrices, Tech. Report MPI-I-2006-1-006,
+            Saarbrucken, 2006
+    """
+
+    # handle the trivial cases
+    if M.rows == 0 and M.cols == 0:
+        return M._new(1, 1, [M.one])
+    elif M.rows == 1 and M.cols == 1:
+        return M._new(2, 1, [M.one, -M[0,0]])
+
+    submat, toeplitz = _berkowitz_toeplitz_matrix(M)
+
+    return toeplitz.multiply(_berkowitz_vector(submat), dotprodsimp=None)
+
+
+def _adjugate(M, method="berkowitz"):
+    """Returns the adjugate, or classical adjoint, of
+    a matrix.  That is, the transpose of the matrix of cofactors.
+
+    https://en.wikipedia.org/wiki/Adjugate
+
+    Parameters
+    ==========
+
+    method : string, optional
+        Method to use to find the cofactors, can be "bareiss", "berkowitz" or
+        "lu".
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2], [3, 4]])
+    >>> M.adjugate()
+    Matrix([
+    [ 4, -2],
+    [-3,  1]])
+
+    See Also
+    ========
+
+    cofactor_matrix
+    sympy.matrices.common.MatrixCommon.transpose
+    """
+
+    return M.cofactor_matrix(method=method).transpose()
+
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _charpoly(M, x='lambda', simplify=_simplify):
+    """Computes characteristic polynomial det(x*I - M) where I is
+    the identity matrix.
+
+    A PurePoly is returned, so using different variables for ``x`` does
+    not affect the comparison or the polynomials:
+
+    Parameters
+    ==========
+
+    x : string, optional
+        Name for the "lambda" variable, defaults to "lambda".
+
+    simplify : function, optional
+        Simplification function to use on the characteristic polynomial
+        calculated. Defaults to ``simplify``.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> from sympy.abc import x, y
+    >>> M = Matrix([[1, 3], [2, 0]])
+    >>> M.charpoly()
+    PurePoly(lambda**2 - lambda - 6, lambda, domain='ZZ')
+    >>> M.charpoly(x) == M.charpoly(y)
+    True
+    >>> M.charpoly(x) == M.charpoly(y)
+    True
+
+    Specifying ``x`` is optional; a symbol named ``lambda`` is used by
+    default (which looks good when pretty-printed in unicode):
+
+    >>> M.charpoly().as_expr()
+    lambda**2 - lambda - 6
+
+    And if ``x`` clashes with an existing symbol, underscores will
+    be prepended to the name to make it unique:
+
+    >>> M = Matrix([[1, 2], [x, 0]])
+    >>> M.charpoly(x).as_expr()
+    _x**2 - _x - 2*x
+
+    Whether you pass a symbol or not, the generator can be obtained
+    with the gen attribute since it may not be the same as the symbol
+    that was passed:
+
+    >>> M.charpoly(x).gen
+    _x
+    >>> M.charpoly(x).gen == x
+    False
+
+    Notes
+    =====
+
+    The Samuelson-Berkowitz algorithm is used to compute
+    the characteristic polynomial efficiently and without any
+    division operations.  Thus the characteristic polynomial over any
+    commutative ring without zero divisors can be computed.
+
+    If the determinant det(x*I - M) can be found out easily as
+    in the case of an upper or a lower triangular matrix, then
+    instead of Samuelson-Berkowitz algorithm, eigenvalues are computed
+    and the characteristic polynomial with their help.
+
+    See Also
+    ========
+
+    det
+    """
+
+    if not M.is_square:
+        raise NonSquareMatrixError()
+    if M.is_lower or M.is_upper:
+        diagonal_elements = M.diagonal()
+        x = uniquely_named_symbol(x, diagonal_elements, modify=lambda s: '_' + s)
+        m = 1
+        for i in diagonal_elements:
+            m = m * (x - simplify(i))
+        return PurePoly(m, x)
+
+    berk_vector = _berkowitz_vector(M)
+    x = uniquely_named_symbol(x, berk_vector, modify=lambda s: '_' + s)
+
+    return PurePoly([simplify(a) for a in berk_vector], x)
+
+
+def _cofactor(M, i, j, method="berkowitz"):
+    """Calculate the cofactor of an element.
+
+    Parameters
+    ==========
+
+    method : string, optional
+        Method to use to find the cofactors, can be "bareiss", "berkowitz" or
+        "lu".
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2], [3, 4]])
+    >>> M.cofactor(0, 1)
+    -3
+
+    See Also
+    ========
+
+    cofactor_matrix
+    minor
+    minor_submatrix
+    """
+
+    if not M.is_square or M.rows < 1:
+        raise NonSquareMatrixError()
+
+    return S.NegativeOne**((i + j) % 2) * M.minor(i, j, method)
+
+
+def _cofactor_matrix(M, method="berkowitz"):
+    """Return a matrix containing the cofactor of each element.
+
+    Parameters
+    ==========
+
+    method : string, optional
+        Method to use to find the cofactors, can be "bareiss", "berkowitz" or
+        "lu".
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2], [3, 4]])
+    >>> M.cofactor_matrix()
+    Matrix([
+    [ 4, -3],
+    [-2,  1]])
+
+    See Also
+    ========
+
+    cofactor
+    minor
+    minor_submatrix
+    """
+
+    if not M.is_square or M.rows < 1:
+        raise NonSquareMatrixError()
+
+    return M._new(M.rows, M.cols,
+            lambda i, j: M.cofactor(i, j, method))
+
+def _per(M):
+    """Returns the permanent of a matrix. Unlike determinant,
+    permanent is defined for both square and non-square matrices.
+
+    For an m x n matrix, with m less than or equal to n,
+    it is given as the sum over the permutations s of size
+    less than or equal to m on [1, 2, . . . n] of the product
+    from i = 1 to m of M[i, s[i]]. Taking the transpose will
+    not affect the value of the permanent.
+
+    In the case of a square matrix, this is the same as the permutation
+    definition of the determinant, but it does not take the sign of the
+    permutation into account. Computing the permanent with this definition
+    is quite inefficient, so here the Ryser formula is used.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    >>> M.per()
+    450
+    >>> M = Matrix([1, 5, 7])
+    >>> M.per()
+    13
+
+    References
+    ==========
+
+    .. [1] Prof. Frank Ben's notes: https://math.berkeley.edu/~bernd/ban275.pdf
+    .. [2] Wikipedia article on Permanent: https://en.wikipedia.org/wiki/Permanent_%28mathematics%29
+    .. [3] https://reference.wolfram.com/language/ref/Permanent.html
+    .. [4] Permanent of a rectangular matrix : https://arxiv.org/pdf/0904.3251.pdf
+    """
+    import itertools
+
+    m, n = M.shape
+    if m > n:
+        M = M.T
+        m, n = n, m
+    s = list(range(n))
+
+    subsets = []
+    for i in range(1, m + 1):
+        subsets += list(map(list, itertools.combinations(s, i)))
+
+    perm = 0
+    for subset in subsets:
+        prod = 1
+        sub_len = len(subset)
+        for i in range(m):
+             prod *= sum([M[i, j] for j in subset])
+        perm += prod * S.NegativeOne**sub_len * nC(n - sub_len, m - sub_len)
+    perm *= S.NegativeOne**m
+    return perm.simplify()
+
+def _det_DOM(M):
+    DOM = DomainMatrix.from_Matrix(M, field=True, extension=True)
+    K = DOM.domain
+    return K.to_sympy(DOM.det())
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _det(M, method="bareiss", iszerofunc=None):
+    """Computes the determinant of a matrix if ``M`` is a concrete matrix object
+    otherwise return an expressions ``Determinant(M)`` if ``M`` is a
+    ``MatrixSymbol`` or other expression.
+
+    Parameters
+    ==========
+
+    method : string, optional
+        Specifies the algorithm used for computing the matrix determinant.
+
+        If the matrix is at most 3x3, a hard-coded formula is used and the
+        specified method is ignored. Otherwise, it defaults to
+        ``'bareiss'``.
+
+        Also, if the matrix is an upper or a lower triangular matrix, determinant
+        is computed by simple multiplication of diagonal elements, and the
+        specified method is ignored.
+
+        If it is set to ``'domain-ge'``, then Gaussian elimination method will
+        be used via using DomainMatrix.
+
+        If it is set to ``'bareiss'``, Bareiss' fraction-free algorithm will
+        be used.
+
+        If it is set to ``'berkowitz'``, Berkowitz' algorithm will be used.
+
+        Otherwise, if it is set to ``'lu'``, LU decomposition will be used.
+
+        .. note::
+            For backward compatibility, legacy keys like "bareis" and
+            "det_lu" can still be used to indicate the corresponding
+            methods.
+            And the keys are also case-insensitive for now. However, it is
+            suggested to use the precise keys for specifying the method.
+
+    iszerofunc : FunctionType or None, optional
+        If it is set to ``None``, it will be defaulted to ``_iszero`` if the
+        method is set to ``'bareiss'``, and ``_is_zero_after_expand_mul`` if
+        the method is set to ``'lu'``.
+
+        It can also accept any user-specified zero testing function, if it
+        is formatted as a function which accepts a single symbolic argument
+        and returns ``True`` if it is tested as zero and ``False`` if it
+        tested as non-zero, and also ``None`` if it is undecidable.
+
+    Returns
+    =======
+
+    det : Basic
+        Result of determinant.
+
+    Raises
+    ======
+
+    ValueError
+        If unrecognized keys are given for ``method`` or ``iszerofunc``.
+
+    NonSquareMatrixError
+        If attempted to calculate determinant from a non-square matrix.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, eye, det
+    >>> I3 = eye(3)
+    >>> det(I3)
+    1
+    >>> M = Matrix([[1, 2], [3, 4]])
+    >>> det(M)
+    -2
+    >>> det(M) == M.det()
+    True
+    >>> M.det(method="domain-ge")
+    -2
+    """
+
+    # sanitize `method`
+    method = method.lower()
+
+    if method == "bareis":
+        method = "bareiss"
+    elif method == "det_lu":
+        method = "lu"
+
+    if method not in ("bareiss", "berkowitz", "lu", "domain-ge"):
+        raise ValueError("Determinant method '%s' unrecognized" % method)
+
+    if iszerofunc is None:
+        if method == "bareiss":
+            iszerofunc = _is_zero_after_expand_mul
+        elif method == "lu":
+            iszerofunc = _iszero
+
+    elif not isinstance(iszerofunc, FunctionType):
+        raise ValueError("Zero testing method '%s' unrecognized" % iszerofunc)
+
+    n = M.rows
+
+    if n == M.cols: # square check is done in individual method functions
+        if n == 0:
+            return M.one
+        elif n == 1:
+            return M[0, 0]
+        elif n == 2:
+            m = M[0, 0] * M[1, 1] - M[0, 1] * M[1, 0]
+            return _get_intermediate_simp(_dotprodsimp)(m)
+        elif n == 3:
+            m =  (M[0, 0] * M[1, 1] * M[2, 2]
+                + M[0, 1] * M[1, 2] * M[2, 0]
+                + M[0, 2] * M[1, 0] * M[2, 1]
+                - M[0, 2] * M[1, 1] * M[2, 0]
+                - M[0, 0] * M[1, 2] * M[2, 1]
+                - M[0, 1] * M[1, 0] * M[2, 2])
+            return _get_intermediate_simp(_dotprodsimp)(m)
+
+    dets = []
+    for b in M.strongly_connected_components():
+        if method == "domain-ge": # uses DomainMatrix to evaluate determinant
+            det = _det_DOM(M[b, b])
+        elif method == "bareiss":
+            det = M[b, b]._eval_det_bareiss(iszerofunc=iszerofunc)
+        elif method == "berkowitz":
+            det = M[b, b]._eval_det_berkowitz()
+        elif method == "lu":
+            det = M[b, b]._eval_det_lu(iszerofunc=iszerofunc)
+        dets.append(det)
+    return Mul(*dets)
+
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _det_bareiss(M, iszerofunc=_is_zero_after_expand_mul):
+    """Compute matrix determinant using Bareiss' fraction-free
+    algorithm which is an extension of the well known Gaussian
+    elimination method. This approach is best suited for dense
+    symbolic matrices and will result in a determinant with
+    minimal number of fractions. It means that less term
+    rewriting is needed on resulting formulae.
+
+    Parameters
+    ==========
+
+    iszerofunc : function, optional
+        The function to use to determine zeros when doing an LU decomposition.
+        Defaults to ``lambda x: x.is_zero``.
+
+    TODO: Implement algorithm for sparse matrices (SFF),
+    http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.
+    """
+
+    # Recursively implemented Bareiss' algorithm as per Deanna Richelle Leggett's
+    # thesis http://www.math.usm.edu/perry/Research/Thesis_DRL.pdf
+    def bareiss(mat, cumm=1):
+        if mat.rows == 0:
+            return mat.one
+        elif mat.rows == 1:
+            return mat[0, 0]
+
+        # find a pivot and extract the remaining matrix
+        # With the default iszerofunc, _find_reasonable_pivot slows down
+        # the computation by the factor of 2.5 in one test.
+        # Relevant issues: #10279 and #13877.
+        pivot_pos, pivot_val, _, _ = _find_reasonable_pivot(mat[:, 0], iszerofunc=iszerofunc)
+        if pivot_pos is None:
+            return mat.zero
+
+        # if we have a valid pivot, we'll do a "row swap", so keep the
+        # sign of the det
+        sign = (-1) ** (pivot_pos % 2)
+
+        # we want every row but the pivot row and every column
+        rows = [i for i in range(mat.rows) if i != pivot_pos]
+        cols = list(range(mat.cols))
+        tmp_mat = mat.extract(rows, cols)
+
+        def entry(i, j):
+            ret = (pivot_val*tmp_mat[i, j + 1] - mat[pivot_pos, j + 1]*tmp_mat[i, 0]) / cumm
+            if _get_intermediate_simp_bool(True):
+                return _dotprodsimp(ret)
+            elif not ret.is_Atom:
+                return cancel(ret)
+            return ret
+
+        return sign*bareiss(M._new(mat.rows - 1, mat.cols - 1, entry), pivot_val)
+
+    if not M.is_square:
+        raise NonSquareMatrixError()
+
+    if M.rows == 0:
+        return M.one
+        # sympy/matrices/tests/test_matrices.py contains a test that
+        # suggests that the determinant of a 0 x 0 matrix is one, by
+        # convention.
+
+    return bareiss(M)
+
+
+def _det_berkowitz(M):
+    """ Use the Berkowitz algorithm to compute the determinant."""
+
+    if not M.is_square:
+        raise NonSquareMatrixError()
+
+    if M.rows == 0:
+        return M.one
+        # sympy/matrices/tests/test_matrices.py contains a test that
+        # suggests that the determinant of a 0 x 0 matrix is one, by
+        # convention.
+
+    berk_vector = _berkowitz_vector(M)
+    return (-1)**(len(berk_vector) - 1) * berk_vector[-1]
+
+
+# This functions is a candidate for caching if it gets implemented for matrices.
+def _det_LU(M, iszerofunc=_iszero, simpfunc=None):
+    """ Computes the determinant of a matrix from its LU decomposition.
+    This function uses the LU decomposition computed by
+    LUDecomposition_Simple().
+
+    The keyword arguments iszerofunc and simpfunc are passed to
+    LUDecomposition_Simple().
+    iszerofunc is a callable that returns a boolean indicating if its
+    input is zero, or None if it cannot make the determination.
+    simpfunc is a callable that simplifies its input.
+    The default is simpfunc=None, which indicate that the pivot search
+    algorithm should not attempt to simplify any candidate pivots.
+    If simpfunc fails to simplify its input, then it must return its input
+    instead of a copy.
+
+    Parameters
+    ==========
+
+    iszerofunc : function, optional
+        The function to use to determine zeros when doing an LU decomposition.
+        Defaults to ``lambda x: x.is_zero``.
+
+    simpfunc : function, optional
+        The simplification function to use when looking for zeros for pivots.
+    """
+
+    if not M.is_square:
+        raise NonSquareMatrixError()
+
+    if M.rows == 0:
+        return M.one
+        # sympy/matrices/tests/test_matrices.py contains a test that
+        # suggests that the determinant of a 0 x 0 matrix is one, by
+        # convention.
+
+    lu, row_swaps = M.LUdecomposition_Simple(iszerofunc=iszerofunc,
+            simpfunc=simpfunc)
+    # P*A = L*U => det(A) = det(L)*det(U)/det(P) = det(P)*det(U).
+    # Lower triangular factor L encoded in lu has unit diagonal => det(L) = 1.
+    # P is a permutation matrix => det(P) in {-1, 1} => 1/det(P) = det(P).
+    # LUdecomposition_Simple() returns a list of row exchange index pairs, rather
+    # than a permutation matrix, but det(P) = (-1)**len(row_swaps).
+
+    # Avoid forming the potentially time consuming  product of U's diagonal entries
+    # if the product is zero.
+    # Bottom right entry of U is 0 => det(A) = 0.
+    # It may be impossible to determine if this entry of U is zero when it is symbolic.
+    if iszerofunc(lu[lu.rows-1, lu.rows-1]):
+        return M.zero
+
+    # Compute det(P)
+    det = -M.one if len(row_swaps)%2 else M.one
+
+    # Compute det(U) by calculating the product of U's diagonal entries.
+    # The upper triangular portion of lu is the upper triangular portion of the
+    # U factor in the LU decomposition.
+    for k in range(lu.rows):
+        det *= lu[k, k]
+
+    # return det(P)*det(U)
+    return det
+
+
+def _minor(M, i, j, method="berkowitz"):
+    """Return the (i,j) minor of ``M``.  That is,
+    return the determinant of the matrix obtained by deleting
+    the `i`th row and `j`th column from ``M``.
+
+    Parameters
+    ==========
+
+    i, j : int
+        The row and column to exclude to obtain the submatrix.
+
+    method : string, optional
+        Method to use to find the determinant of the submatrix, can be
+        "bareiss", "berkowitz" or "lu".
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    >>> M.minor(1, 1)
+    -12
+
+    See Also
+    ========
+
+    minor_submatrix
+    cofactor
+    det
+    """
+
+    if not M.is_square:
+        raise NonSquareMatrixError()
+
+    return M.minor_submatrix(i, j).det(method=method)
+
+
+def _minor_submatrix(M, i, j):
+    """Return the submatrix obtained by removing the `i`th row
+    and `j`th column from ``M`` (works with Pythonic negative indices).
+
+    Parameters
+    ==========
+
+    i, j : int
+        The row and column to exclude to obtain the submatrix.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    >>> M.minor_submatrix(1, 1)
+    Matrix([
+    [1, 3],
+    [7, 9]])
+
+    See Also
+    ========
+
+    minor
+    cofactor
+    """
+
+    if i < 0:
+        i += M.rows
+    if j < 0:
+        j += M.cols
+
+    if not 0 <= i < M.rows or not 0 <= j < M.cols:
+        raise ValueError("`i` and `j` must satisfy 0 <= i < ``M.rows`` "
+                            "(%d)" % M.rows + "and 0 <= j < ``M.cols`` (%d)." % M.cols)
+
+    rows = [a for a in range(M.rows) if a != i]
+    cols = [a for a in range(M.cols) if a != j]
+
+    return M.extract(rows, cols)
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@@ -0,0 +1,434 @@
+"""Implementation of the Kronecker product"""
+from functools import reduce
+from math import prod
+
+from sympy.core import Mul, sympify
+from sympy.functions import adjoint
+from sympy.matrices.common import ShapeError
+from sympy.matrices.expressions.matexpr import MatrixExpr
+from sympy.matrices.expressions.transpose import transpose
+from sympy.matrices.expressions.special import Identity
+from sympy.matrices.matrices import MatrixBase
+from sympy.strategies import (
+    canon, condition, distribute, do_one, exhaust, flatten, typed, unpack)
+from sympy.strategies.traverse import bottom_up
+from sympy.utilities import sift
+
+from .matadd import MatAdd
+from .matmul import MatMul
+from .matpow import MatPow
+
+
+def kronecker_product(*matrices):
+    """
+    The Kronecker product of two or more arguments.
+
+    This computes the explicit Kronecker product for subclasses of
+    ``MatrixBase`` i.e. explicit matrices. Otherwise, a symbolic
+    ``KroneckerProduct`` object is returned.
+
+
+    Examples
+    ========
+
+    For ``MatrixSymbol`` arguments a ``KroneckerProduct`` object is returned.
+    Elements of this matrix can be obtained by indexing, or for MatrixSymbols
+    with known dimension the explicit matrix can be obtained with
+    ``.as_explicit()``
+
+    >>> from sympy import kronecker_product, MatrixSymbol
+    >>> A = MatrixSymbol('A', 2, 2)
+    >>> B = MatrixSymbol('B', 2, 2)
+    >>> kronecker_product(A)
+    A
+    >>> kronecker_product(A, B)
+    KroneckerProduct(A, B)
+    >>> kronecker_product(A, B)[0, 1]
+    A[0, 0]*B[0, 1]
+    >>> kronecker_product(A, B).as_explicit()
+    Matrix([
+        [A[0, 0]*B[0, 0], A[0, 0]*B[0, 1], A[0, 1]*B[0, 0], A[0, 1]*B[0, 1]],
+        [A[0, 0]*B[1, 0], A[0, 0]*B[1, 1], A[0, 1]*B[1, 0], A[0, 1]*B[1, 1]],
+        [A[1, 0]*B[0, 0], A[1, 0]*B[0, 1], A[1, 1]*B[0, 0], A[1, 1]*B[0, 1]],
+        [A[1, 0]*B[1, 0], A[1, 0]*B[1, 1], A[1, 1]*B[1, 0], A[1, 1]*B[1, 1]]])
+
+    For explicit matrices the Kronecker product is returned as a Matrix
+
+    >>> from sympy import Matrix, kronecker_product
+    >>> sigma_x = Matrix([
+    ... [0, 1],
+    ... [1, 0]])
+    ...
+    >>> Isigma_y = Matrix([
+    ... [0, 1],
+    ... [-1, 0]])
+    ...
+    >>> kronecker_product(sigma_x, Isigma_y)
+    Matrix([
+    [ 0, 0,  0, 1],
+    [ 0, 0, -1, 0],
+    [ 0, 1,  0, 0],
+    [-1, 0,  0, 0]])
+
+    See Also
+    ========
+        KroneckerProduct
+
+    """
+    if not matrices:
+        raise TypeError("Empty Kronecker product is undefined")
+    if len(matrices) == 1:
+        return matrices[0]
+    else:
+        return KroneckerProduct(*matrices).doit()
+
+
+class KroneckerProduct(MatrixExpr):
+    """
+    The Kronecker product of two or more arguments.
+
+    The Kronecker product is a non-commutative product of matrices.
+    Given two matrices of dimension (m, n) and (s, t) it produces a matrix
+    of dimension (m s, n t).
+
+    This is a symbolic object that simply stores its argument without
+    evaluating it. To actually compute the product, use the function
+    ``kronecker_product()`` or call the ``.doit()`` or  ``.as_explicit()``
+    methods.
+
+    >>> from sympy import KroneckerProduct, MatrixSymbol
+    >>> A = MatrixSymbol('A', 5, 5)
+    >>> B = MatrixSymbol('B', 5, 5)
+    >>> isinstance(KroneckerProduct(A, B), KroneckerProduct)
+    True
+    """
+    is_KroneckerProduct = True
+
+    def __new__(cls, *args, check=True):
+        args = list(map(sympify, args))
+        if all(a.is_Identity for a in args):
+            ret = Identity(prod(a.rows for a in args))
+            if all(isinstance(a, MatrixBase) for a in args):
+                return ret.as_explicit()
+            else:
+                return ret
+
+        if check:
+            validate(*args)
+        return super().__new__(cls, *args)
+
+    @property
+    def shape(self):
+        rows, cols = self.args[0].shape
+        for mat in self.args[1:]:
+            rows *= mat.rows
+            cols *= mat.cols
+        return (rows, cols)
+
+    def _entry(self, i, j, **kwargs):
+        result = 1
+        for mat in reversed(self.args):
+            i, m = divmod(i, mat.rows)
+            j, n = divmod(j, mat.cols)
+            result *= mat[m, n]
+        return result
+
+    def _eval_adjoint(self):
+        return KroneckerProduct(*list(map(adjoint, self.args))).doit()
+
+    def _eval_conjugate(self):
+        return KroneckerProduct(*[a.conjugate() for a in self.args]).doit()
+
+    def _eval_transpose(self):
+        return KroneckerProduct(*list(map(transpose, self.args))).doit()
+
+    def _eval_trace(self):
+        from .trace import trace
+        return Mul(*[trace(a) for a in self.args])
+
+    def _eval_determinant(self):
+        from .determinant import det, Determinant
+        if not all(a.is_square for a in self.args):
+            return Determinant(self)
+
+        m = self.rows
+        return Mul(*[det(a)**(m/a.rows) for a in self.args])
+
+    def _eval_inverse(self):
+        try:
+            return KroneckerProduct(*[a.inverse() for a in self.args])
+        except ShapeError:
+            from sympy.matrices.expressions.inverse import Inverse
+            return Inverse(self)
+
+    def structurally_equal(self, other):
+        '''Determine whether two matrices have the same Kronecker product structure
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerProduct, MatrixSymbol, symbols
+        >>> m, n = symbols(r'm, n', integer=True)
+        >>> A = MatrixSymbol('A', m, m)
+        >>> B = MatrixSymbol('B', n, n)
+        >>> C = MatrixSymbol('C', m, m)
+        >>> D = MatrixSymbol('D', n, n)
+        >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(C, D))
+        True
+        >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(D, C))
+        False
+        >>> KroneckerProduct(A, B).structurally_equal(C)
+        False
+        '''
+        # Inspired by BlockMatrix
+        return (isinstance(other, KroneckerProduct)
+                and self.shape == other.shape
+                and len(self.args) == len(other.args)
+                and all(a.shape == b.shape for (a, b) in zip(self.args, other.args)))
+
+    def has_matching_shape(self, other):
+        '''Determine whether two matrices have the appropriate structure to bring matrix
+        multiplication inside the KroneckerProdut
+
+        Examples
+        ========
+        >>> from sympy import KroneckerProduct, MatrixSymbol, symbols
+        >>> m, n = symbols(r'm, n', integer=True)
+        >>> A = MatrixSymbol('A', m, n)
+        >>> B = MatrixSymbol('B', n, m)
+        >>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(B, A))
+        True
+        >>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(A, B))
+        False
+        >>> KroneckerProduct(A, B).has_matching_shape(A)
+        False
+        '''
+        return (isinstance(other, KroneckerProduct)
+                and self.cols == other.rows
+                and len(self.args) == len(other.args)
+                and all(a.cols == b.rows for (a, b) in zip(self.args, other.args)))
+
+    def _eval_expand_kroneckerproduct(self, **hints):
+        return flatten(canon(typed({KroneckerProduct: distribute(KroneckerProduct, MatAdd)}))(self))
+
+    def _kronecker_add(self, other):
+        if self.structurally_equal(other):
+            return self.__class__(*[a + b for (a, b) in zip(self.args, other.args)])
+        else:
+            return self + other
+
+    def _kronecker_mul(self, other):
+        if self.has_matching_shape(other):
+            return self.__class__(*[a*b for (a, b) in zip(self.args, other.args)])
+        else:
+            return self * other
+
+    def doit(self, **hints):
+        deep = hints.get('deep', True)
+        if deep:
+            args = [arg.doit(**hints) for arg in self.args]
+        else:
+            args = self.args
+        return canonicalize(KroneckerProduct(*args))
+
+
+def validate(*args):
+    if not all(arg.is_Matrix for arg in args):
+        raise TypeError("Mix of Matrix and Scalar symbols")
+
+
+# rules
+
+def extract_commutative(kron):
+    c_part = []
+    nc_part = []
+    for arg in kron.args:
+        c, nc = arg.args_cnc()
+        c_part.extend(c)
+        nc_part.append(Mul._from_args(nc))
+
+    c_part = Mul(*c_part)
+    if c_part != 1:
+        return c_part*KroneckerProduct(*nc_part)
+    return kron
+
+
+def matrix_kronecker_product(*matrices):
+    """Compute the Kronecker product of a sequence of SymPy Matrices.
+
+    This is the standard Kronecker product of matrices [1].
+
+    Parameters
+    ==========
+
+    matrices : tuple of MatrixBase instances
+        The matrices to take the Kronecker product of.
+
+    Returns
+    =======
+
+    matrix : MatrixBase
+        The Kronecker product matrix.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> from sympy.matrices.expressions.kronecker import (
+    ... matrix_kronecker_product)
+
+    >>> m1 = Matrix([[1,2],[3,4]])
+    >>> m2 = Matrix([[1,0],[0,1]])
+    >>> matrix_kronecker_product(m1, m2)
+    Matrix([
+    [1, 0, 2, 0],
+    [0, 1, 0, 2],
+    [3, 0, 4, 0],
+    [0, 3, 0, 4]])
+    >>> matrix_kronecker_product(m2, m1)
+    Matrix([
+    [1, 2, 0, 0],
+    [3, 4, 0, 0],
+    [0, 0, 1, 2],
+    [0, 0, 3, 4]])
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Kronecker_product
+    """
+    # Make sure we have a sequence of Matrices
+    if not all(isinstance(m, MatrixBase) for m in matrices):
+        raise TypeError(
+            'Sequence of Matrices expected, got: %s' % repr(matrices)
+        )
+
+    # Pull out the first element in the product.
+    matrix_expansion = matrices[-1]
+    # Do the kronecker product working from right to left.
+    for mat in reversed(matrices[:-1]):
+        rows = mat.rows
+        cols = mat.cols
+        # Go through each row appending kronecker product to.
+        # running matrix_expansion.
+        for i in range(rows):
+            start = matrix_expansion*mat[i*cols]
+            # Go through each column joining each item
+            for j in range(cols - 1):
+                start = start.row_join(
+                    matrix_expansion*mat[i*cols + j + 1]
+                )
+            # If this is the first element, make it the start of the
+            # new row.
+            if i == 0:
+                next = start
+            else:
+                next = next.col_join(start)
+        matrix_expansion = next
+
+    MatrixClass = max(matrices, key=lambda M: M._class_priority).__class__
+    if isinstance(matrix_expansion, MatrixClass):
+        return matrix_expansion
+    else:
+        return MatrixClass(matrix_expansion)
+
+
+def explicit_kronecker_product(kron):
+    # Make sure we have a sequence of Matrices
+    if not all(isinstance(m, MatrixBase) for m in kron.args):
+        return kron
+
+    return matrix_kronecker_product(*kron.args)
+
+
+rules = (unpack,
+         explicit_kronecker_product,
+         flatten,
+         extract_commutative)
+
+canonicalize = exhaust(condition(lambda x: isinstance(x, KroneckerProduct),
+                                 do_one(*rules)))
+
+
+def _kronecker_dims_key(expr):
+    if isinstance(expr, KroneckerProduct):
+        return tuple(a.shape for a in expr.args)
+    else:
+        return (0,)
+
+
+def kronecker_mat_add(expr):
+    args = sift(expr.args, _kronecker_dims_key)
+    nonkrons = args.pop((0,), None)
+    if not args:
+        return expr
+
+    krons = [reduce(lambda x, y: x._kronecker_add(y), group)
+             for group in args.values()]
+
+    if not nonkrons:
+        return MatAdd(*krons)
+    else:
+        return MatAdd(*krons) + nonkrons
+
+
+def kronecker_mat_mul(expr):
+    # modified from block matrix code
+    factor, matrices = expr.as_coeff_matrices()
+
+    i = 0
+    while i < len(matrices) - 1:
+        A, B = matrices[i:i+2]
+        if isinstance(A, KroneckerProduct) and isinstance(B, KroneckerProduct):
+            matrices[i] = A._kronecker_mul(B)
+            matrices.pop(i+1)
+        else:
+            i += 1
+
+    return factor*MatMul(*matrices)
+
+
+def kronecker_mat_pow(expr):
+    if isinstance(expr.base, KroneckerProduct) and all(a.is_square for a in expr.base.args):
+        return KroneckerProduct(*[MatPow(a, expr.exp) for a in expr.base.args])
+    else:
+        return expr
+
+
+def combine_kronecker(expr):
+    """Combine KronekeckerProduct with expression.
+
+    If possible write operations on KroneckerProducts of compatible shapes
+    as a single KroneckerProduct.
+
+    Examples
+    ========
+
+    >>> from sympy.matrices.expressions import combine_kronecker
+    >>> from sympy import MatrixSymbol, KroneckerProduct, symbols
+    >>> m, n = symbols(r'm, n', integer=True)
+    >>> A = MatrixSymbol('A', m, n)
+    >>> B = MatrixSymbol('B', n, m)
+    >>> combine_kronecker(KroneckerProduct(A, B)*KroneckerProduct(B, A))
+    KroneckerProduct(A*B, B*A)
+    >>> combine_kronecker(KroneckerProduct(A, B)+KroneckerProduct(B.T, A.T))
+    KroneckerProduct(A + B.T, B + A.T)
+    >>> C = MatrixSymbol('C', n, n)
+    >>> D = MatrixSymbol('D', m, m)
+    >>> combine_kronecker(KroneckerProduct(C, D)**m)
+    KroneckerProduct(C**m, D**m)
+    """
+    def haskron(expr):
+        return isinstance(expr, MatrixExpr) and expr.has(KroneckerProduct)
+
+    rule = exhaust(
+        bottom_up(exhaust(condition(haskron, typed(
+            {MatAdd: kronecker_mat_add,
+             MatMul: kronecker_mat_mul,
+             MatPow: kronecker_mat_pow})))))
+    result = rule(expr)
+    doit = getattr(result, 'doit', None)
+    if doit is not None:
+        return doit()
+    else:
+        return result
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--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/repmatrix.py
@@ -0,0 +1,759 @@
+from collections import defaultdict
+
+from operator import index as index_
+
+from sympy.core.expr import Expr
+from sympy.core.kind import Kind, NumberKind, UndefinedKind
+from sympy.core.numbers import Integer, Rational
+from sympy.core.sympify import _sympify, SympifyError
+from sympy.core.singleton import S
+from sympy.polys.domains import ZZ, QQ, EXRAW
+from sympy.polys.matrices import DomainMatrix
+from sympy.utilities.exceptions import sympy_deprecation_warning
+from sympy.utilities.iterables import is_sequence
+from sympy.utilities.misc import filldedent
+
+from .common import classof
+from .matrices import MatrixBase, MatrixKind, ShapeError
+
+
+class RepMatrix(MatrixBase):
+    """Matrix implementation based on DomainMatrix as an internal representation.
+
+    The RepMatrix class is a superclass for Matrix, ImmutableMatrix,
+    SparseMatrix and ImmutableSparseMatrix which are the main usable matrix
+    classes in SymPy. Most methods on this class are simply forwarded to
+    DomainMatrix.
+    """
+
+    #
+    # MatrixBase is the common superclass for all of the usable explicit matrix
+    # classes in SymPy. The idea is that MatrixBase is an abstract class though
+    # and that subclasses will implement the lower-level methods.
+    #
+    # RepMatrix is a subclass of MatrixBase that uses DomainMatrix as an
+    # internal representation and delegates lower-level methods to
+    # DomainMatrix. All of SymPy's standard explicit matrix classes subclass
+    # RepMatrix and so use DomainMatrix internally.
+    #
+    # A RepMatrix uses an internal DomainMatrix with the domain set to ZZ, QQ
+    # or EXRAW. The EXRAW domain is equivalent to the previous implementation
+    # of Matrix that used Expr for the elements. The ZZ and QQ domains are used
+    # when applicable just because they are compatible with the previous
+    # implementation but are much more efficient. Other domains such as QQ[x]
+    # are not used because they differ from Expr in some way (e.g. automatic
+    # expansion of powers and products).
+    #
+
+    _rep: DomainMatrix
+
+    def __eq__(self, other):
+        # Skip sympify for mutable matrices...
+        if not isinstance(other, RepMatrix):
+            try:
+                other = _sympify(other)
+            except SympifyError:
+                return NotImplemented
+            if not isinstance(other, RepMatrix):
+                return NotImplemented
+
+        return self._rep.unify_eq(other._rep)
+
+    @classmethod
+    def _unify_element_sympy(cls, rep, element):
+        domain = rep.domain
+        element = _sympify(element)
+
+        if domain != EXRAW:
+            # The domain can only be ZZ, QQ or EXRAW
+            if element.is_Integer:
+                new_domain = domain
+            elif element.is_Rational:
+                new_domain = QQ
+            else:
+                new_domain = EXRAW
+
+            # XXX: This converts the domain for all elements in the matrix
+            # which can be slow. This happens e.g. if __setitem__ changes one
+            # element to something that does not fit in the domain
+            if new_domain != domain:
+                rep = rep.convert_to(new_domain)
+                domain = new_domain
+
+            if domain != EXRAW:
+                element = new_domain.from_sympy(element)
+
+        if domain == EXRAW and not isinstance(element, Expr):
+            sympy_deprecation_warning(
+                """
+                non-Expr objects in a Matrix is deprecated. Matrix represents
+                a mathematical matrix. To represent a container of non-numeric
+                entities, Use a list of lists, TableForm, NumPy array, or some
+                other data structure instead.
+                """,
+                deprecated_since_version="1.9",
+                active_deprecations_target="deprecated-non-expr-in-matrix",
+                stacklevel=4,
+            )
+
+        return rep, element
+
+    @classmethod
+    def _dod_to_DomainMatrix(cls, rows, cols, dod, types):
+
+        if not all(issubclass(typ, Expr) for typ in types):
+            sympy_deprecation_warning(
+                """
+                non-Expr objects in a Matrix is deprecated. Matrix represents
+                a mathematical matrix. To represent a container of non-numeric
+                entities, Use a list of lists, TableForm, NumPy array, or some
+                other data structure instead.
+                """,
+                deprecated_since_version="1.9",
+                active_deprecations_target="deprecated-non-expr-in-matrix",
+                stacklevel=6,
+            )
+
+        rep = DomainMatrix(dod, (rows, cols), EXRAW)
+
+        if all(issubclass(typ, Rational) for typ in types):
+            if all(issubclass(typ, Integer) for typ in types):
+                rep = rep.convert_to(ZZ)
+            else:
+                rep = rep.convert_to(QQ)
+
+        return rep
+
+    @classmethod
+    def _flat_list_to_DomainMatrix(cls, rows, cols, flat_list):
+
+        elements_dod = defaultdict(dict)
+        for n, element in enumerate(flat_list):
+            if element != 0:
+                i, j = divmod(n, cols)
+                elements_dod[i][j] = element
+
+        types = set(map(type, flat_list))
+
+        rep = cls._dod_to_DomainMatrix(rows, cols, elements_dod, types)
+        return rep
+
+    @classmethod
+    def _smat_to_DomainMatrix(cls, rows, cols, smat):
+
+        elements_dod = defaultdict(dict)
+        for (i, j), element in smat.items():
+            if element != 0:
+                elements_dod[i][j] = element
+
+        types = set(map(type, smat.values()))
+
+        rep = cls._dod_to_DomainMatrix(rows, cols, elements_dod, types)
+        return rep
+
+    def flat(self):
+        return self._rep.to_sympy().to_list_flat()
+
+    def _eval_tolist(self):
+        return self._rep.to_sympy().to_list()
+
+    def _eval_todok(self):
+        return self._rep.to_sympy().to_dok()
+
+    def _eval_values(self):
+        return list(self.todok().values())
+
+    def copy(self):
+        return self._fromrep(self._rep.copy())
+
+    @property
+    def kind(self) -> MatrixKind:
+        domain = self._rep.domain
+        element_kind: Kind
+        if domain in (ZZ, QQ):
+            element_kind = NumberKind
+        elif domain == EXRAW:
+            kinds = {e.kind for e in self.values()}
+            if len(kinds) == 1:
+                [element_kind] = kinds
+            else:
+                element_kind = UndefinedKind
+        else: # pragma: no cover
+            raise RuntimeError("Domain should only be ZZ, QQ or EXRAW")
+        return MatrixKind(element_kind)
+
+    def _eval_has(self, *patterns):
+        # if the matrix has any zeros, see if S.Zero
+        # has the pattern.  If _smat is full length,
+        # the matrix has no zeros.
+        zhas = False
+        dok = self.todok()
+        if len(dok) != self.rows*self.cols:
+            zhas = S.Zero.has(*patterns)
+        return zhas or any(value.has(*patterns) for value in dok.values())
+
+    def _eval_is_Identity(self):
+        if not all(self[i, i] == 1 for i in range(self.rows)):
+            return False
+        return len(self.todok()) == self.rows
+
+    def _eval_is_symmetric(self, simpfunc):
+        diff = (self - self.T).applyfunc(simpfunc)
+        return len(diff.values()) == 0
+
+    def _eval_transpose(self):
+        """Returns the transposed SparseMatrix of this SparseMatrix.
+
+        Examples
+        ========
+
+        >>> from sympy import SparseMatrix
+        >>> a = SparseMatrix(((1, 2), (3, 4)))
+        >>> a
+        Matrix([
+        [1, 2],
+        [3, 4]])
+        >>> a.T
+        Matrix([
+        [1, 3],
+        [2, 4]])
+        """
+        return self._fromrep(self._rep.transpose())
+
+    def _eval_col_join(self, other):
+        return self._fromrep(self._rep.vstack(other._rep))
+
+    def _eval_row_join(self, other):
+        return self._fromrep(self._rep.hstack(other._rep))
+
+    def _eval_extract(self, rowsList, colsList):
+        return self._fromrep(self._rep.extract(rowsList, colsList))
+
+    def __getitem__(self, key):
+        return _getitem_RepMatrix(self, key)
+
+    @classmethod
+    def _eval_zeros(cls, rows, cols):
+        rep = DomainMatrix.zeros((rows, cols), ZZ)
+        return cls._fromrep(rep)
+
+    @classmethod
+    def _eval_eye(cls, rows, cols):
+        rep = DomainMatrix.eye((rows, cols), ZZ)
+        return cls._fromrep(rep)
+
+    def _eval_add(self, other):
+        return classof(self, other)._fromrep(self._rep + other._rep)
+
+    def _eval_matrix_mul(self, other):
+        return classof(self, other)._fromrep(self._rep * other._rep)
+
+    def _eval_matrix_mul_elementwise(self, other):
+        selfrep, otherrep = self._rep.unify(other._rep)
+        newrep = selfrep.mul_elementwise(otherrep)
+        return classof(self, other)._fromrep(newrep)
+
+    def _eval_scalar_mul(self, other):
+        rep, other = self._unify_element_sympy(self._rep, other)
+        return self._fromrep(rep.scalarmul(other))
+
+    def _eval_scalar_rmul(self, other):
+        rep, other = self._unify_element_sympy(self._rep, other)
+        return self._fromrep(rep.rscalarmul(other))
+
+    def _eval_Abs(self):
+        return self._fromrep(self._rep.applyfunc(abs))
+
+    def _eval_conjugate(self):
+        rep = self._rep
+        domain = rep.domain
+        if domain in (ZZ, QQ):
+            return self.copy()
+        else:
+            return self._fromrep(rep.applyfunc(lambda e: e.conjugate()))
+
+    def equals(self, other, failing_expression=False):
+        """Applies ``equals`` to corresponding elements of the matrices,
+        trying to prove that the elements are equivalent, returning True
+        if they are, False if any pair is not, and None (or the first
+        failing expression if failing_expression is True) if it cannot
+        be decided if the expressions are equivalent or not. This is, in
+        general, an expensive operation.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> from sympy.abc import x
+        >>> A = Matrix([x*(x - 1), 0])
+        >>> B = Matrix([x**2 - x, 0])
+        >>> A == B
+        False
+        >>> A.simplify() == B.simplify()
+        True
+        >>> A.equals(B)
+        True
+        >>> A.equals(2)
+        False
+
+        See Also
+        ========
+        sympy.core.expr.Expr.equals
+        """
+        if self.shape != getattr(other, 'shape', None):
+            return False
+
+        rv = True
+        for i in range(self.rows):
+            for j in range(self.cols):
+                ans = self[i, j].equals(other[i, j], failing_expression)
+                if ans is False:
+                    return False
+                elif ans is not True and rv is True:
+                    rv = ans
+        return rv
+
+
+class MutableRepMatrix(RepMatrix):
+    """Mutable matrix based on DomainMatrix as the internal representation"""
+
+    #
+    # MutableRepMatrix is a subclass of RepMatrix that adds/overrides methods
+    # to make the instances mutable. MutableRepMatrix is a superclass for both
+    # MutableDenseMatrix and MutableSparseMatrix.
+    #
+
+    is_zero = False
+
+    def __new__(cls, *args, **kwargs):
+        return cls._new(*args, **kwargs)
+
+    @classmethod
+    def _new(cls, *args, copy=True, **kwargs):
+        if copy is False:
+            # The input was rows, cols, [list].
+            # It should be used directly without creating a copy.
+            if len(args) != 3:
+                raise TypeError("'copy=False' requires a matrix be initialized as rows,cols,[list]")
+            rows, cols, flat_list = args
+        else:
+            rows, cols, flat_list = cls._handle_creation_inputs(*args, **kwargs)
+            flat_list = list(flat_list) # create a shallow copy
+
+        rep = cls._flat_list_to_DomainMatrix(rows, cols, flat_list)
+
+        return cls._fromrep(rep)
+
+    @classmethod
+    def _fromrep(cls, rep):
+        obj = super().__new__(cls)
+        obj.rows, obj.cols = rep.shape
+        obj._rep = rep
+        return obj
+
+    def copy(self):
+        return self._fromrep(self._rep.copy())
+
+    def as_mutable(self):
+        return self.copy()
+
+    def __setitem__(self, key, value):
+        """
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, I, zeros, ones
+        >>> m = Matrix(((1, 2+I), (3, 4)))
+        >>> m
+        Matrix([
+        [1, 2 + I],
+        [3,     4]])
+        >>> m[1, 0] = 9
+        >>> m
+        Matrix([
+        [1, 2 + I],
+        [9,     4]])
+        >>> m[1, 0] = [[0, 1]]
+
+        To replace row r you assign to position r*m where m
+        is the number of columns:
+
+        >>> M = zeros(4)
+        >>> m = M.cols
+        >>> M[3*m] = ones(1, m)*2; M
+        Matrix([
+        [0, 0, 0, 0],
+        [0, 0, 0, 0],
+        [0, 0, 0, 0],
+        [2, 2, 2, 2]])
+
+        And to replace column c you can assign to position c:
+
+        >>> M[2] = ones(m, 1)*4; M
+        Matrix([
+        [0, 0, 4, 0],
+        [0, 0, 4, 0],
+        [0, 0, 4, 0],
+        [2, 2, 4, 2]])
+        """
+        rv = self._setitem(key, value)
+        if rv is not None:
+            i, j, value = rv
+            self._rep, value = self._unify_element_sympy(self._rep, value)
+            self._rep.rep.setitem(i, j, value)
+
+    def _eval_col_del(self, col):
+        self._rep = DomainMatrix.hstack(self._rep[:,:col], self._rep[:,col+1:])
+        self.cols -= 1
+
+    def _eval_row_del(self, row):
+        self._rep = DomainMatrix.vstack(self._rep[:row,:], self._rep[row+1:, :])
+        self.rows -= 1
+
+    def _eval_col_insert(self, col, other):
+        other = self._new(other)
+        return self.hstack(self[:,:col], other, self[:,col:])
+
+    def _eval_row_insert(self, row, other):
+        other = self._new(other)
+        return self.vstack(self[:row,:], other, self[row:,:])
+
+    def col_op(self, j, f):
+        """In-place operation on col j using two-arg functor whose args are
+        interpreted as (self[i, j], i).
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> M = eye(3)
+        >>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
+        Matrix([
+        [1, 2, 0],
+        [0, 1, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+        col
+        row_op
+        """
+        for i in range(self.rows):
+            self[i, j] = f(self[i, j], i)
+
+    def col_swap(self, i, j):
+        """Swap the two given columns of the matrix in-place.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> M = Matrix([[1, 0], [1, 0]])
+        >>> M
+        Matrix([
+        [1, 0],
+        [1, 0]])
+        >>> M.col_swap(0, 1)
+        >>> M
+        Matrix([
+        [0, 1],
+        [0, 1]])
+
+        See Also
+        ========
+
+        col
+        row_swap
+        """
+        for k in range(0, self.rows):
+            self[k, i], self[k, j] = self[k, j], self[k, i]
+
+    def row_op(self, i, f):
+        """In-place operation on row ``i`` using two-arg functor whose args are
+        interpreted as ``(self[i, j], j)``.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> M = eye(3)
+        >>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
+        Matrix([
+        [1, 0, 0],
+        [2, 1, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+        row
+        zip_row_op
+        col_op
+
+        """
+        for j in range(self.cols):
+            self[i, j] = f(self[i, j], j)
+
+    def row_swap(self, i, j):
+        """Swap the two given rows of the matrix in-place.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> M = Matrix([[0, 1], [1, 0]])
+        >>> M
+        Matrix([
+        [0, 1],
+        [1, 0]])
+        >>> M.row_swap(0, 1)
+        >>> M
+        Matrix([
+        [1, 0],
+        [0, 1]])
+
+        See Also
+        ========
+
+        row
+        col_swap
+        """
+        for k in range(0, self.cols):
+            self[i, k], self[j, k] = self[j, k], self[i, k]
+
+    def zip_row_op(self, i, k, f):
+        """In-place operation on row ``i`` using two-arg functor whose args are
+        interpreted as ``(self[i, j], self[k, j])``.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> M = eye(3)
+        >>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
+        Matrix([
+        [1, 0, 0],
+        [2, 1, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+        row
+        row_op
+        col_op
+
+        """
+        for j in range(self.cols):
+            self[i, j] = f(self[i, j], self[k, j])
+
+    def copyin_list(self, key, value):
+        """Copy in elements from a list.
+
+        Parameters
+        ==========
+
+        key : slice
+            The section of this matrix to replace.
+        value : iterable
+            The iterable to copy values from.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> I = eye(3)
+        >>> I[:2, 0] = [1, 2] # col
+        >>> I
+        Matrix([
+        [1, 0, 0],
+        [2, 1, 0],
+        [0, 0, 1]])
+        >>> I[1, :2] = [[3, 4]]
+        >>> I
+        Matrix([
+        [1, 0, 0],
+        [3, 4, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+
+        copyin_matrix
+        """
+        if not is_sequence(value):
+            raise TypeError("`value` must be an ordered iterable, not %s." % type(value))
+        return self.copyin_matrix(key, type(self)(value))
+
+    def copyin_matrix(self, key, value):
+        """Copy in values from a matrix into the given bounds.
+
+        Parameters
+        ==========
+
+        key : slice
+            The section of this matrix to replace.
+        value : Matrix
+            The matrix to copy values from.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, eye
+        >>> M = Matrix([[0, 1], [2, 3], [4, 5]])
+        >>> I = eye(3)
+        >>> I[:3, :2] = M
+        >>> I
+        Matrix([
+        [0, 1, 0],
+        [2, 3, 0],
+        [4, 5, 1]])
+        >>> I[0, 1] = M
+        >>> I
+        Matrix([
+        [0, 0, 1],
+        [2, 2, 3],
+        [4, 4, 5]])
+
+        See Also
+        ========
+
+        copyin_list
+        """
+        rlo, rhi, clo, chi = self.key2bounds(key)
+        shape = value.shape
+        dr, dc = rhi - rlo, chi - clo
+        if shape != (dr, dc):
+            raise ShapeError(filldedent("The Matrix `value` doesn't have the "
+                                        "same dimensions "
+                                        "as the in sub-Matrix given by `key`."))
+
+        for i in range(value.rows):
+            for j in range(value.cols):
+                self[i + rlo, j + clo] = value[i, j]
+
+    def fill(self, value):
+        """Fill self with the given value.
+
+        Notes
+        =====
+
+        Unless many values are going to be deleted (i.e. set to zero)
+        this will create a matrix that is slower than a dense matrix in
+        operations.
+
+        Examples
+        ========
+
+        >>> from sympy import SparseMatrix
+        >>> M = SparseMatrix.zeros(3); M
+        Matrix([
+        [0, 0, 0],
+        [0, 0, 0],
+        [0, 0, 0]])
+        >>> M.fill(1); M
+        Matrix([
+        [1, 1, 1],
+        [1, 1, 1],
+        [1, 1, 1]])
+
+        See Also
+        ========
+
+        zeros
+        ones
+        """
+        value = _sympify(value)
+        if not value:
+            self._rep = DomainMatrix.zeros(self.shape, EXRAW)
+        else:
+            elements_dod = {i: {j: value for j in range(self.cols)} for i in range(self.rows)}
+            self._rep = DomainMatrix(elements_dod, self.shape, EXRAW)
+
+
+def _getitem_RepMatrix(self, key):
+    """Return portion of self defined by key. If the key involves a slice
+    then a list will be returned (if key is a single slice) or a matrix
+    (if key was a tuple involving a slice).
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, I
+    >>> m = Matrix([
+    ... [1, 2 + I],
+    ... [3, 4    ]])
+
+    If the key is a tuple that does not involve a slice then that element
+    is returned:
+
+    >>> m[1, 0]
+    3
+
+    When a tuple key involves a slice, a matrix is returned. Here, the
+    first column is selected (all rows, column 0):
+
+    >>> m[:, 0]
+    Matrix([
+    [1],
+    [3]])
+
+    If the slice is not a tuple then it selects from the underlying
+    list of elements that are arranged in row order and a list is
+    returned if a slice is involved:
+
+    >>> m[0]
+    1
+    >>> m[::2]
+    [1, 3]
+    """
+    if isinstance(key, tuple):
+        i, j = key
+        try:
+            return self._rep.getitem_sympy(index_(i), index_(j))
+        except (TypeError, IndexError):
+            if (isinstance(i, Expr) and not i.is_number) or (isinstance(j, Expr) and not j.is_number):
+                if ((j < 0) is True) or ((j >= self.shape[1]) is True) or\
+                   ((i < 0) is True) or ((i >= self.shape[0]) is True):
+                    raise ValueError("index out of boundary")
+                from sympy.matrices.expressions.matexpr import MatrixElement
+                return MatrixElement(self, i, j)
+
+            if isinstance(i, slice):
+                i = range(self.rows)[i]
+            elif is_sequence(i):
+                pass
+            else:
+                i = [i]
+            if isinstance(j, slice):
+                j = range(self.cols)[j]
+            elif is_sequence(j):
+                pass
+            else:
+                j = [j]
+            return self.extract(i, j)
+
+    else:
+        # Index/slice like a flattened list
+        rows, cols = self.shape
+
+        # Raise the appropriate exception:
+        if not rows * cols:
+            return [][key]
+
+        rep = self._rep.rep
+        domain = rep.domain
+        is_slice = isinstance(key, slice)
+
+        if is_slice:
+            values = [rep.getitem(*divmod(n, cols)) for n in range(rows * cols)[key]]
+        else:
+            values = [rep.getitem(*divmod(index_(key), cols))]
+
+        if domain != EXRAW:
+            to_sympy = domain.to_sympy
+            values = [to_sympy(val) for val in values]
+
+        if is_slice:
+            return values
+        else:
+            return values[0]
diff --git a/llmeval-env/lib/python3.10/site-packages/sympy/matrices/solvers.py b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/solvers.py
new file mode 100644
index 0000000000000000000000000000000000000000..713ea1052398568beb09e4628ca69491076c36ae
--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/solvers.py
@@ -0,0 +1,862 @@
+from sympy.core.function import expand_mul
+from sympy.core.symbol import Dummy, uniquely_named_symbol, symbols
+from sympy.utilities.iterables import numbered_symbols
+
+from .common import ShapeError, NonSquareMatrixError, NonInvertibleMatrixError
+from .eigen import _fuzzy_positive_definite
+from .utilities import _get_intermediate_simp, _iszero
+
+
+def _diagonal_solve(M, rhs):
+    """Solves ``Ax = B`` efficiently, where A is a diagonal Matrix,
+    with non-zero diagonal entries.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, eye
+    >>> A = eye(2)*2
+    >>> B = Matrix([[1, 2], [3, 4]])
+    >>> A.diagonal_solve(B) == B/2
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    if not M.is_diagonal():
+        raise TypeError("Matrix should be diagonal")
+    if rhs.rows != M.rows:
+        raise TypeError("Size mismatch")
+
+    return M._new(
+        rhs.rows, rhs.cols, lambda i, j: rhs[i, j] / M[i, i])
+
+
+def _lower_triangular_solve(M, rhs):
+    """Solves ``Ax = B``, where A is a lower triangular matrix.
+
+    See Also
+    ========
+
+    upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if rhs.rows != M.rows:
+        raise ShapeError("Matrices size mismatch.")
+    if not M.is_lower:
+        raise ValueError("Matrix must be lower triangular.")
+
+    dps = _get_intermediate_simp()
+    X   = MutableDenseMatrix.zeros(M.rows, rhs.cols)
+
+    for j in range(rhs.cols):
+        for i in range(M.rows):
+            if M[i, i] == 0:
+                raise TypeError("Matrix must be non-singular.")
+
+            X[i, j] = dps((rhs[i, j] - sum(M[i, k]*X[k, j]
+                                        for k in range(i))) / M[i, i])
+
+    return M._new(X)
+
+def _lower_triangular_solve_sparse(M, rhs):
+    """Solves ``Ax = B``, where A is a lower triangular matrix.
+
+    See Also
+    ========
+
+    upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if rhs.rows != M.rows:
+        raise ShapeError("Matrices size mismatch.")
+    if not M.is_lower:
+        raise ValueError("Matrix must be lower triangular.")
+
+    dps  = _get_intermediate_simp()
+    rows = [[] for i in range(M.rows)]
+
+    for i, j, v in M.row_list():
+        if i > j:
+            rows[i].append((j, v))
+
+    X = rhs.as_mutable()
+
+    for j in range(rhs.cols):
+        for i in range(rhs.rows):
+            for u, v in rows[i]:
+                X[i, j] -= v*X[u, j]
+
+            X[i, j] = dps(X[i, j] / M[i, i])
+
+    return M._new(X)
+
+
+def _upper_triangular_solve(M, rhs):
+    """Solves ``Ax = B``, where A is an upper triangular matrix.
+
+    See Also
+    ========
+
+    lower_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if rhs.rows != M.rows:
+        raise ShapeError("Matrix size mismatch.")
+    if not M.is_upper:
+        raise TypeError("Matrix is not upper triangular.")
+
+    dps = _get_intermediate_simp()
+    X   = MutableDenseMatrix.zeros(M.rows, rhs.cols)
+
+    for j in range(rhs.cols):
+        for i in reversed(range(M.rows)):
+            if M[i, i] == 0:
+                raise ValueError("Matrix must be non-singular.")
+
+            X[i, j] = dps((rhs[i, j] - sum(M[i, k]*X[k, j]
+                                        for k in range(i + 1, M.rows))) / M[i, i])
+
+    return M._new(X)
+
+def _upper_triangular_solve_sparse(M, rhs):
+    """Solves ``Ax = B``, where A is an upper triangular matrix.
+
+    See Also
+    ========
+
+    lower_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if rhs.rows != M.rows:
+        raise ShapeError("Matrix size mismatch.")
+    if not M.is_upper:
+        raise TypeError("Matrix is not upper triangular.")
+
+    dps  = _get_intermediate_simp()
+    rows = [[] for i in range(M.rows)]
+
+    for i, j, v in M.row_list():
+        if i < j:
+            rows[i].append((j, v))
+
+    X = rhs.as_mutable()
+
+    for j in range(rhs.cols):
+        for i in reversed(range(rhs.rows)):
+            for u, v in reversed(rows[i]):
+                X[i, j] -= v*X[u, j]
+
+            X[i, j] = dps(X[i, j] / M[i, i])
+
+    return M._new(X)
+
+
+def _cholesky_solve(M, rhs):
+    """Solves ``Ax = B`` using Cholesky decomposition,
+    for a general square non-singular matrix.
+    For a non-square matrix with rows > cols,
+    the least squares solution is returned.
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    if M.rows < M.cols:
+        raise NotImplementedError(
+            'Under-determined System. Try M.gauss_jordan_solve(rhs)')
+
+    hermitian = True
+    reform    = False
+
+    if M.is_symmetric():
+        hermitian = False
+    elif not M.is_hermitian:
+        reform = True
+
+    if reform or _fuzzy_positive_definite(M) is False:
+        H         = M.H
+        M         = H.multiply(M)
+        rhs       = H.multiply(rhs)
+        hermitian = not M.is_symmetric()
+
+    L = M.cholesky(hermitian=hermitian)
+    Y = L.lower_triangular_solve(rhs)
+
+    if hermitian:
+        return (L.H).upper_triangular_solve(Y)
+    else:
+        return (L.T).upper_triangular_solve(Y)
+
+
+def _LDLsolve(M, rhs):
+    """Solves ``Ax = B`` using LDL decomposition,
+    for a general square and non-singular matrix.
+
+    For a non-square matrix with rows > cols,
+    the least squares solution is returned.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, eye
+    >>> A = eye(2)*2
+    >>> B = Matrix([[1, 2], [3, 4]])
+    >>> A.LDLsolve(B) == B/2
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.LDLdecomposition
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LUsolve
+    QRsolve
+    pinv_solve
+    """
+
+    if M.rows < M.cols:
+        raise NotImplementedError(
+            'Under-determined System. Try M.gauss_jordan_solve(rhs)')
+
+    hermitian = True
+    reform    = False
+
+    if M.is_symmetric():
+        hermitian = False
+    elif not M.is_hermitian:
+        reform = True
+
+    if reform or _fuzzy_positive_definite(M) is False:
+        H         = M.H
+        M         = H.multiply(M)
+        rhs       = H.multiply(rhs)
+        hermitian = not M.is_symmetric()
+
+    L, D = M.LDLdecomposition(hermitian=hermitian)
+    Y    = L.lower_triangular_solve(rhs)
+    Z    = D.diagonal_solve(Y)
+
+    if hermitian:
+        return (L.H).upper_triangular_solve(Z)
+    else:
+        return (L.T).upper_triangular_solve(Z)
+
+
+def _LUsolve(M, rhs, iszerofunc=_iszero):
+    """Solve the linear system ``Ax = rhs`` for ``x`` where ``A = M``.
+
+    This is for symbolic matrices, for real or complex ones use
+    mpmath.lu_solve or mpmath.qr_solve.
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    QRsolve
+    pinv_solve
+    LUdecomposition
+    """
+
+    if rhs.rows != M.rows:
+        raise ShapeError(
+            "``M`` and ``rhs`` must have the same number of rows.")
+
+    m = M.rows
+    n = M.cols
+
+    if m < n:
+        raise NotImplementedError("Underdetermined systems not supported.")
+
+    try:
+        A, perm = M.LUdecomposition_Simple(
+            iszerofunc=_iszero, rankcheck=True)
+    except ValueError:
+        raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
+
+    dps = _get_intermediate_simp()
+    b   = rhs.permute_rows(perm).as_mutable()
+
+    # forward substitution, all diag entries are scaled to 1
+    for i in range(m):
+        for j in range(min(i, n)):
+            scale = A[i, j]
+            b.zip_row_op(i, j, lambda x, y: dps(x - y * scale))
+
+    # consistency check for overdetermined systems
+    if m > n:
+        for i in range(n, m):
+            for j in range(b.cols):
+                if not iszerofunc(b[i, j]):
+                    raise ValueError("The system is inconsistent.")
+
+        b = b[0:n, :]   # truncate zero rows if consistent
+
+    # backward substitution
+    for i in range(n - 1, -1, -1):
+        for j in range(i + 1, n):
+            scale = A[i, j]
+            b.zip_row_op(i, j, lambda x, y: dps(x - y * scale))
+
+        scale = A[i, i]
+        b.row_op(i, lambda x, _: dps(x / scale))
+
+    return rhs.__class__(b)
+
+
+def _QRsolve(M, b):
+    """Solve the linear system ``Ax = b``.
+
+    ``M`` is the matrix ``A``, the method argument is the vector
+    ``b``.  The method returns the solution vector ``x``.  If ``b`` is a
+    matrix, the system is solved for each column of ``b`` and the
+    return value is a matrix of the same shape as ``b``.
+
+    This method is slower (approximately by a factor of 2) but
+    more stable for floating-point arithmetic than the LUsolve method.
+    However, LUsolve usually uses an exact arithmetic, so you do not need
+    to use QRsolve.
+
+    This is mainly for educational purposes and symbolic matrices, for real
+    (or complex) matrices use mpmath.qr_solve.
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    pinv_solve
+    QRdecomposition
+    """
+
+    dps  = _get_intermediate_simp(expand_mul, expand_mul)
+    Q, R = M.QRdecomposition()
+    y    = Q.T * b
+
+    # back substitution to solve R*x = y:
+    # We build up the result "backwards" in the vector 'x' and reverse it
+    # only in the end.
+    x = []
+    n = R.rows
+
+    for j in range(n - 1, -1, -1):
+        tmp = y[j, :]
+
+        for k in range(j + 1, n):
+            tmp -= R[j, k] * x[n - 1 - k]
+
+        tmp = dps(tmp)
+
+        x.append(tmp / R[j, j])
+
+    return M.vstack(*x[::-1])
+
+
+def _gauss_jordan_solve(M, B, freevar=False):
+    """
+    Solves ``Ax = B`` using Gauss Jordan elimination.
+
+    There may be zero, one, or infinite solutions.  If one solution
+    exists, it will be returned. If infinite solutions exist, it will
+    be returned parametrically. If no solutions exist, It will throw
+    ValueError.
+
+    Parameters
+    ==========
+
+    B : Matrix
+        The right hand side of the equation to be solved for.  Must have
+        the same number of rows as matrix A.
+
+    freevar : boolean, optional
+        Flag, when set to `True` will return the indices of the free
+        variables in the solutions (column Matrix), for a system that is
+        undetermined (e.g. A has more columns than rows), for which
+        infinite solutions are possible, in terms of arbitrary
+        values of free variables. Default `False`.
+
+    Returns
+    =======
+
+    x : Matrix
+        The matrix that will satisfy ``Ax = B``.  Will have as many rows as
+        matrix A has columns, and as many columns as matrix B.
+
+    params : Matrix
+        If the system is underdetermined (e.g. A has more columns than
+        rows), infinite solutions are possible, in terms of arbitrary
+        parameters. These arbitrary parameters are returned as params
+        Matrix.
+
+    free_var_index : List, optional
+        If the system is underdetermined (e.g. A has more columns than
+        rows), infinite solutions are possible, in terms of arbitrary
+        values of free variables. Then the indices of the free variables
+        in the solutions (column Matrix) are returned by free_var_index,
+        if the flag `freevar` is set to `True`.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([[1, 2, 1, 1], [1, 2, 2, -1], [2, 4, 0, 6]])
+    >>> B = Matrix([7, 12, 4])
+    >>> sol, params = A.gauss_jordan_solve(B)
+    >>> sol
+    Matrix([
+    [-2*tau0 - 3*tau1 + 2],
+    [                 tau0],
+    [           2*tau1 + 5],
+    [                 tau1]])
+    >>> params
+    Matrix([
+    [tau0],
+    [tau1]])
+    >>> taus_zeroes = { tau:0 for tau in params }
+    >>> sol_unique = sol.xreplace(taus_zeroes)
+    >>> sol_unique
+        Matrix([
+    [2],
+    [0],
+    [5],
+    [0]])
+
+
+    >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
+    >>> B = Matrix([3, 6, 9])
+    >>> sol, params = A.gauss_jordan_solve(B)
+    >>> sol
+    Matrix([
+    [-1],
+    [ 2],
+    [ 0]])
+    >>> params
+    Matrix(0, 1, [])
+
+    >>> A = Matrix([[2, -7], [-1, 4]])
+    >>> B = Matrix([[-21, 3], [12, -2]])
+    >>> sol, params = A.gauss_jordan_solve(B)
+    >>> sol
+    Matrix([
+    [0, -2],
+    [3, -1]])
+    >>> params
+    Matrix(0, 2, [])
+
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([[1, 2, 1, 1], [1, 2, 2, -1], [2, 4, 0, 6]])
+    >>> B = Matrix([7, 12, 4])
+    >>> sol, params, freevars = A.gauss_jordan_solve(B, freevar=True)
+    >>> sol
+    Matrix([
+    [-2*tau0 - 3*tau1 + 2],
+    [                 tau0],
+    [           2*tau1 + 5],
+    [                 tau1]])
+    >>> params
+    Matrix([
+    [tau0],
+    [tau1]])
+    >>> freevars
+    [1, 3]
+
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gaussian_elimination
+
+    """
+
+    from sympy.matrices import Matrix, zeros
+
+    cls      = M.__class__
+    aug      = M.hstack(M.copy(), B.copy())
+    B_cols   = B.cols
+    row, col = aug[:, :-B_cols].shape
+
+    # solve by reduced row echelon form
+    A, pivots = aug.rref(simplify=True)
+    A, v      = A[:, :-B_cols], A[:, -B_cols:]
+    pivots    = list(filter(lambda p: p < col, pivots))
+    rank      = len(pivots)
+
+    # Get index of free symbols (free parameters)
+    # non-pivots columns are free variables
+    free_var_index = [c for c in range(A.cols) if c not in pivots]
+
+    # Bring to block form
+    permutation = Matrix(pivots + free_var_index).T
+
+    # check for existence of solutions
+    # rank of aug Matrix should be equal to rank of coefficient matrix
+    if not v[rank:, :].is_zero_matrix:
+        raise ValueError("Linear system has no solution")
+
+    # Free parameters
+    # what are current unnumbered free symbol names?
+    name = uniquely_named_symbol('tau', aug,
+            compare=lambda i: str(i).rstrip('1234567890'),
+            modify=lambda s: '_' + s).name
+    gen  = numbered_symbols(name)
+    tau  = Matrix([next(gen) for k in range((col - rank)*B_cols)]).reshape(
+            col - rank, B_cols)
+
+    # Full parametric solution
+    V        = A[:rank, free_var_index]
+    vt       = v[:rank, :]
+    free_sol = tau.vstack(vt - V * tau, tau)
+
+    # Undo permutation
+    sol = zeros(col, B_cols)
+
+    for k in range(col):
+        sol[permutation[k], :] = free_sol[k,:]
+
+    sol, tau = cls(sol), cls(tau)
+
+    if freevar:
+        return sol, tau, free_var_index
+    else:
+        return sol, tau
+
+
+def _pinv_solve(M, B, arbitrary_matrix=None):
+    """Solve ``Ax = B`` using the Moore-Penrose pseudoinverse.
+
+    There may be zero, one, or infinite solutions.  If one solution
+    exists, it will be returned.  If infinite solutions exist, one will
+    be returned based on the value of arbitrary_matrix.  If no solutions
+    exist, the least-squares solution is returned.
+
+    Parameters
+    ==========
+
+    B : Matrix
+        The right hand side of the equation to be solved for.  Must have
+        the same number of rows as matrix A.
+    arbitrary_matrix : Matrix
+        If the system is underdetermined (e.g. A has more columns than
+        rows), infinite solutions are possible, in terms of an arbitrary
+        matrix.  This parameter may be set to a specific matrix to use
+        for that purpose; if so, it must be the same shape as x, with as
+        many rows as matrix A has columns, and as many columns as matrix
+        B.  If left as None, an appropriate matrix containing dummy
+        symbols in the form of ``wn_m`` will be used, with n and m being
+        row and column position of each symbol.
+
+    Returns
+    =======
+
+    x : Matrix
+        The matrix that will satisfy ``Ax = B``.  Will have as many rows as
+        matrix A has columns, and as many columns as matrix B.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([[1, 2, 3], [4, 5, 6]])
+    >>> B = Matrix([7, 8])
+    >>> A.pinv_solve(B)
+    Matrix([
+    [ _w0_0/6 - _w1_0/3 + _w2_0/6 - 55/18],
+    [-_w0_0/3 + 2*_w1_0/3 - _w2_0/3 + 1/9],
+    [ _w0_0/6 - _w1_0/3 + _w2_0/6 + 59/18]])
+    >>> A.pinv_solve(B, arbitrary_matrix=Matrix([0, 0, 0]))
+    Matrix([
+    [-55/18],
+    [   1/9],
+    [ 59/18]])
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.lower_triangular_solve
+    sympy.matrices.dense.DenseMatrix.upper_triangular_solve
+    gauss_jordan_solve
+    cholesky_solve
+    diagonal_solve
+    LDLsolve
+    LUsolve
+    QRsolve
+    pinv
+
+    Notes
+    =====
+
+    This may return either exact solutions or least squares solutions.
+    To determine which, check ``A * A.pinv() * B == B``.  It will be
+    True if exact solutions exist, and False if only a least-squares
+    solution exists.  Be aware that the left hand side of that equation
+    may need to be simplified to correctly compare to the right hand
+    side.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#Obtaining_all_solutions_of_a_linear_system
+
+    """
+
+    from sympy.matrices import eye
+
+    A      = M
+    A_pinv = M.pinv()
+
+    if arbitrary_matrix is None:
+        rows, cols       = A.cols, B.cols
+        w                = symbols('w:{}_:{}'.format(rows, cols), cls=Dummy)
+        arbitrary_matrix = M.__class__(cols, rows, w).T
+
+    return A_pinv.multiply(B) + (eye(A.cols) -
+            A_pinv.multiply(A)).multiply(arbitrary_matrix)
+
+
+def _solve(M, rhs, method='GJ'):
+    """Solves linear equation where the unique solution exists.
+
+    Parameters
+    ==========
+
+    rhs : Matrix
+        Vector representing the right hand side of the linear equation.
+
+    method : string, optional
+        If set to ``'GJ'`` or ``'GE'``, the Gauss-Jordan elimination will be
+        used, which is implemented in the routine ``gauss_jordan_solve``.
+
+        If set to ``'LU'``, ``LUsolve`` routine will be used.
+
+        If set to ``'QR'``, ``QRsolve`` routine will be used.
+
+        If set to ``'PINV'``, ``pinv_solve`` routine will be used.
+
+        It also supports the methods available for special linear systems
+
+        For positive definite systems:
+
+        If set to ``'CH'``, ``cholesky_solve`` routine will be used.
+
+        If set to ``'LDL'``, ``LDLsolve`` routine will be used.
+
+        To use a different method and to compute the solution via the
+        inverse, use a method defined in the .inv() docstring.
+
+    Returns
+    =======
+
+    solutions : Matrix
+        Vector representing the solution.
+
+    Raises
+    ======
+
+    ValueError
+        If there is not a unique solution then a ``ValueError`` will be
+        raised.
+
+        If ``M`` is not square, a ``ValueError`` and a different routine
+        for solving the system will be suggested.
+    """
+
+    if method in ('GJ', 'GE'):
+        try:
+            soln, param = M.gauss_jordan_solve(rhs)
+
+            if param:
+                raise NonInvertibleMatrixError("Matrix det == 0; not invertible. "
+                "Try ``M.gauss_jordan_solve(rhs)`` to obtain a parametric solution.")
+
+        except ValueError:
+            raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
+
+        return soln
+
+    elif method == 'LU':
+        return M.LUsolve(rhs)
+    elif method == 'CH':
+        return M.cholesky_solve(rhs)
+    elif method == 'QR':
+        return M.QRsolve(rhs)
+    elif method == 'LDL':
+        return M.LDLsolve(rhs)
+    elif method == 'PINV':
+        return M.pinv_solve(rhs)
+    else:
+        return M.inv(method=method).multiply(rhs)
+
+
+def _solve_least_squares(M, rhs, method='CH'):
+    """Return the least-square fit to the data.
+
+    Parameters
+    ==========
+
+    rhs : Matrix
+        Vector representing the right hand side of the linear equation.
+
+    method : string or boolean, optional
+        If set to ``'CH'``, ``cholesky_solve`` routine will be used.
+
+        If set to ``'LDL'``, ``LDLsolve`` routine will be used.
+
+        If set to ``'QR'``, ``QRsolve`` routine will be used.
+
+        If set to ``'PINV'``, ``pinv_solve`` routine will be used.
+
+        Otherwise, the conjugate of ``M`` will be used to create a system
+        of equations that is passed to ``solve`` along with the hint
+        defined by ``method``.
+
+    Returns
+    =======
+
+    solutions : Matrix
+        Vector representing the solution.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, ones
+    >>> A = Matrix([1, 2, 3])
+    >>> B = Matrix([2, 3, 4])
+    >>> S = Matrix(A.row_join(B))
+    >>> S
+    Matrix([
+    [1, 2],
+    [2, 3],
+    [3, 4]])
+
+    If each line of S represent coefficients of Ax + By
+    and x and y are [2, 3] then S*xy is:
+
+    >>> r = S*Matrix([2, 3]); r
+    Matrix([
+    [ 8],
+    [13],
+    [18]])
+
+    But let's add 1 to the middle value and then solve for the
+    least-squares value of xy:
+
+    >>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy
+    Matrix([
+    [ 5/3],
+    [10/3]])
+
+    The error is given by S*xy - r:
+
+    >>> S*xy - r
+    Matrix([
+    [1/3],
+    [1/3],
+    [1/3]])
+    >>> _.norm().n(2)
+    0.58
+
+    If a different xy is used, the norm will be higher:
+
+    >>> xy += ones(2, 1)/10
+    >>> (S*xy - r).norm().n(2)
+    1.5
+
+    """
+
+    if method == 'CH':
+        return M.cholesky_solve(rhs)
+    elif method == 'QR':
+        return M.QRsolve(rhs)
+    elif method == 'LDL':
+        return M.LDLsolve(rhs)
+    elif method == 'PINV':
+        return M.pinv_solve(rhs)
+    else:
+        t = M.H
+        return (t * M).solve(t * rhs, method=method)
diff --git a/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_determinant.py b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_determinant.py
new file mode 100644
index 0000000000000000000000000000000000000000..7690e57bd61031d4b6e8f5e4f2e147bbe8d92273
--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_determinant.py
@@ -0,0 +1,414 @@
+import random
+from sympy.core.numbers import I
+from sympy.core.numbers import Rational
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.polys.polytools import Poly
+from sympy.matrices import Matrix, eye, ones
+from sympy.abc import x, y, z
+from sympy.testing.pytest import raises
+from sympy.matrices.common import NonSquareMatrixError
+from sympy.functions.combinatorial.factorials import factorial, subfactorial
+
+
+def test_determinant():
+    M = Matrix()
+
+    assert M.det() == 1
+    # Evaluating these directly because they are never reached via M.det()
+    assert M._eval_det_bareiss() == 1
+    assert M._eval_det_berkowitz() == 1
+    assert M._eval_det_lu() == 1
+
+    M = Matrix([ [0] ])
+
+    assert M.det() == 0
+    assert M._eval_det_bareiss() == 0
+    assert M._eval_det_berkowitz() == 0
+    assert M._eval_det_lu() == 0
+
+    M = Matrix([ [5] ])
+
+    assert M.det() == 5
+    assert M._eval_det_bareiss() == 5
+    assert M._eval_det_berkowitz() == 5
+    assert M._eval_det_lu() == 5
+
+    M = Matrix(( (-3,  2),
+                 ( 8, -5) ))
+
+    assert M.det(method="domain-ge") == -1
+    assert M.det(method="bareiss") == -1
+    assert M.det(method="berkowitz") == -1
+    assert M.det(method="lu") == -1
+
+    M = Matrix(( (x,   1),
+                 (y, 2*y) ))
+
+    assert M.det(method="domain-ge") == 2*x*y - y
+    assert M.det(method="bareiss") == 2*x*y - y
+    assert M.det(method="berkowitz") == 2*x*y - y
+    assert M.det(method="lu") == 2*x*y - y
+
+    M = Matrix(( (1, 1, 1),
+                 (1, 2, 3),
+                 (1, 3, 6) ))
+
+    assert M.det(method="domain-ge") == 1
+    assert M.det(method="bareiss") == 1
+    assert M.det(method="berkowitz") == 1
+    assert M.det(method="lu") == 1
+
+    M = Matrix(( ( 3, -2,  0, 5),
+                 (-2,  1, -2, 2),
+                 ( 0, -2,  5, 0),
+                 ( 5,  0,  3, 4) ))
+
+    assert M.det(method="domain-ge") == -289
+    assert M.det(method="bareiss") == -289
+    assert M.det(method="berkowitz") == -289
+    assert M.det(method="lu") == -289
+
+    M = Matrix(( ( 1,  2,  3,  4),
+                 ( 5,  6,  7,  8),
+                 ( 9, 10, 11, 12),
+                 (13, 14, 15, 16) ))
+
+    assert M.det(method="domain-ge") == 0
+    assert M.det(method="bareiss") == 0
+    assert M.det(method="berkowitz") == 0
+    assert M.det(method="lu") == 0
+
+    M = Matrix(( (3, 2, 0, 0, 0),
+                 (0, 3, 2, 0, 0),
+                 (0, 0, 3, 2, 0),
+                 (0, 0, 0, 3, 2),
+                 (2, 0, 0, 0, 3) ))
+
+    assert M.det(method="domain-ge") == 275
+    assert M.det(method="bareiss") == 275
+    assert M.det(method="berkowitz") == 275
+    assert M.det(method="lu") == 275
+
+    M = Matrix(( ( 3,  0,  0, 0),
+                 (-2,  1,  0, 0),
+                 ( 0, -2,  5, 0),
+                 ( 5,  0,  3, 4) ))
+
+    assert M.det(method="domain-ge") == 60
+    assert M.det(method="bareiss") == 60
+    assert M.det(method="berkowitz") == 60
+    assert M.det(method="lu") == 60
+
+    M = Matrix(( ( 1,  0,  0,  0),
+                 ( 5,  0,  0,  0),
+                 ( 9, 10, 11, 0),
+                 (13, 14, 15, 16) ))
+
+    assert M.det(method="domain-ge") == 0
+    assert M.det(method="bareiss") == 0
+    assert M.det(method="berkowitz") == 0
+    assert M.det(method="lu") == 0
+
+    M = Matrix(( (3, 2, 0, 0, 0),
+                 (0, 3, 2, 0, 0),
+                 (0, 0, 3, 2, 0),
+                 (0, 0, 0, 3, 2),
+                 (0, 0, 0, 0, 3) ))
+
+    assert M.det(method="domain-ge") == 243
+    assert M.det(method="bareiss") == 243
+    assert M.det(method="berkowitz") == 243
+    assert M.det(method="lu") == 243
+
+    M = Matrix(( (1, 0,  1,  2, 12),
+                 (2, 0,  1,  1,  4),
+                 (2, 1,  1, -1,  3),
+                 (3, 2, -1,  1,  8),
+                 (1, 1,  1,  0,  6) ))
+
+    assert M.det(method="domain-ge") == -55
+    assert M.det(method="bareiss") == -55
+    assert M.det(method="berkowitz") == -55
+    assert M.det(method="lu") == -55
+
+    M = Matrix(( (-5,  2,  3,  4,  5),
+                 ( 1, -4,  3,  4,  5),
+                 ( 1,  2, -3,  4,  5),
+                 ( 1,  2,  3, -2,  5),
+                 ( 1,  2,  3,  4, -1) ))
+
+    assert M.det(method="domain-ge") == 11664
+    assert M.det(method="bareiss") == 11664
+    assert M.det(method="berkowitz") == 11664
+    assert M.det(method="lu") == 11664
+
+    M = Matrix(( ( 2,  7, -1, 3, 2),
+                 ( 0,  0,  1, 0, 1),
+                 (-2,  0,  7, 0, 2),
+                 (-3, -2,  4, 5, 3),
+                 ( 1,  0,  0, 0, 1) ))
+
+    assert M.det(method="domain-ge") == 123
+    assert M.det(method="bareiss") == 123
+    assert M.det(method="berkowitz") == 123
+    assert M.det(method="lu") == 123
+
+    M = Matrix(( (x, y, z),
+                 (1, 0, 0),
+                 (y, z, x) ))
+
+    assert M.det(method="domain-ge") == z**2 - x*y
+    assert M.det(method="bareiss") == z**2 - x*y
+    assert M.det(method="berkowitz") == z**2 - x*y
+    assert M.det(method="lu") == z**2 - x*y
+
+    # issue 13835
+    a = symbols('a')
+    M = lambda n: Matrix([[i + a*j for i in range(n)]
+                          for j in range(n)])
+    assert M(5).det() == 0
+    assert M(6).det() == 0
+    assert M(7).det() == 0
+
+def test_issue_14517():
+    M = Matrix([
+        [   0, 10*I,    10*I,       0],
+        [10*I,    0,       0,    10*I],
+        [10*I,    0, 5 + 2*I,    10*I],
+        [   0, 10*I,    10*I, 5 + 2*I]])
+    ev = M.eigenvals()
+    # test one random eigenvalue, the computation is a little slow
+    test_ev = random.choice(list(ev.keys()))
+    assert (M - test_ev*eye(4)).det() == 0
+
+def test_legacy_det():
+    # Minimal support for legacy keys for 'method' in det()
+    # Partially copied from test_determinant()
+
+    M = Matrix(( ( 3, -2,  0, 5),
+                 (-2,  1, -2, 2),
+                 ( 0, -2,  5, 0),
+                 ( 5,  0,  3, 4) ))
+
+    assert M.det(method="bareis") == -289
+    assert M.det(method="det_lu") == -289
+    assert M.det(method="det_LU") == -289
+
+    M = Matrix(( (3, 2, 0, 0, 0),
+                 (0, 3, 2, 0, 0),
+                 (0, 0, 3, 2, 0),
+                 (0, 0, 0, 3, 2),
+                 (2, 0, 0, 0, 3) ))
+
+    assert M.det(method="bareis") == 275
+    assert M.det(method="det_lu") == 275
+    assert M.det(method="Bareis") == 275
+
+    M = Matrix(( (1, 0,  1,  2, 12),
+                 (2, 0,  1,  1,  4),
+                 (2, 1,  1, -1,  3),
+                 (3, 2, -1,  1,  8),
+                 (1, 1,  1,  0,  6) ))
+
+    assert M.det(method="bareis") == -55
+    assert M.det(method="det_lu") == -55
+    assert M.det(method="BAREISS") == -55
+
+    M = Matrix(( ( 3,  0,  0, 0),
+                 (-2,  1,  0, 0),
+                 ( 0, -2,  5, 0),
+                 ( 5,  0,  3, 4) ))
+
+    assert M.det(method="bareiss") == 60
+    assert M.det(method="berkowitz") == 60
+    assert M.det(method="lu") == 60
+
+    M = Matrix(( ( 1,  0,  0,  0),
+                 ( 5,  0,  0,  0),
+                 ( 9, 10, 11, 0),
+                 (13, 14, 15, 16) ))
+
+    assert M.det(method="bareiss") == 0
+    assert M.det(method="berkowitz") == 0
+    assert M.det(method="lu") == 0
+
+    M = Matrix(( (3, 2, 0, 0, 0),
+                 (0, 3, 2, 0, 0),
+                 (0, 0, 3, 2, 0),
+                 (0, 0, 0, 3, 2),
+                 (0, 0, 0, 0, 3) ))
+
+    assert M.det(method="bareiss") == 243
+    assert M.det(method="berkowitz") == 243
+    assert M.det(method="lu") == 243
+
+    M = Matrix(( (-5,  2,  3,  4,  5),
+                 ( 1, -4,  3,  4,  5),
+                 ( 1,  2, -3,  4,  5),
+                 ( 1,  2,  3, -2,  5),
+                 ( 1,  2,  3,  4, -1) ))
+
+    assert M.det(method="bareis") == 11664
+    assert M.det(method="det_lu") == 11664
+    assert M.det(method="BERKOWITZ") == 11664
+
+    M = Matrix(( ( 2,  7, -1, 3, 2),
+                 ( 0,  0,  1, 0, 1),
+                 (-2,  0,  7, 0, 2),
+                 (-3, -2,  4, 5, 3),
+                 ( 1,  0,  0, 0, 1) ))
+
+    assert M.det(method="bareis") == 123
+    assert M.det(method="det_lu") == 123
+    assert M.det(method="LU") == 123
+
+def eye_Determinant(n):
+    return Matrix(n, n, lambda i, j: int(i == j))
+
+def zeros_Determinant(n):
+    return Matrix(n, n, lambda i, j: 0)
+
+def test_det():
+    a = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
+    raises(NonSquareMatrixError, lambda: a.det())
+
+    z = zeros_Determinant(2)
+    ey = eye_Determinant(2)
+    assert z.det() == 0
+    assert ey.det() == 1
+
+    x = Symbol('x')
+    a = Matrix(0, 0, [])
+    b = Matrix(1, 1, [5])
+    c = Matrix(2, 2, [1, 2, 3, 4])
+    d = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
+    e = Matrix(4, 4,
+        [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
+    from sympy.abc import i, j, k, l, m, n
+    f = Matrix(3, 3, [i, l, m, 0, j, n, 0, 0, k])
+    g = Matrix(3, 3, [i, 0, 0, l, j, 0, m, n, k])
+    h = Matrix(3, 3, [x**3, 0, 0, i, x**-1, 0, j, k, x**-2])
+    # the method keyword for `det` doesn't kick in until 4x4 matrices,
+    # so there is no need to test all methods on smaller ones
+
+    assert a.det() == 1
+    assert b.det() == 5
+    assert c.det() == -2
+    assert d.det() == 3
+    assert e.det() == 4*x - 24
+    assert e.det(method="domain-ge") == 4*x - 24
+    assert e.det(method='bareiss') == 4*x - 24
+    assert e.det(method='berkowitz') == 4*x - 24
+    assert f.det() == i*j*k
+    assert g.det() == i*j*k
+    assert h.det() == 1
+    raises(ValueError, lambda: e.det(iszerofunc="test"))
+
+def test_permanent():
+    M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    assert M.per() == 450
+    for i in range(1, 12):
+        assert ones(i, i).per() == ones(i, i).T.per() == factorial(i)
+        assert (ones(i, i)-eye(i)).per() == (ones(i, i)-eye(i)).T.per() == subfactorial(i)
+
+    a1, a2, a3, a4, a5 = symbols('a_1 a_2 a_3 a_4 a_5')
+    M = Matrix([a1, a2, a3, a4, a5])
+    assert M.per() == M.T.per() == a1 + a2 + a3 + a4 + a5
+
+def test_adjugate():
+    x = Symbol('x')
+    e = Matrix(4, 4,
+        [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
+
+    adj = Matrix([
+        [   4,         -8,         4,         0],
+        [  76, -14*x - 68,  14*x - 8, -4*x + 24],
+        [-122, 17*x + 142, -21*x + 4,  8*x - 48],
+        [  48,  -4*x - 72,       8*x, -4*x + 24]])
+    assert e.adjugate() == adj
+    assert e.adjugate(method='bareiss') == adj
+    assert e.adjugate(method='berkowitz') == adj
+
+    a = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
+    raises(NonSquareMatrixError, lambda: a.adjugate())
+
+def test_util():
+    R = Rational
+
+    v1 = Matrix(1, 3, [1, 2, 3])
+    v2 = Matrix(1, 3, [3, 4, 5])
+    assert v1.norm() == sqrt(14)
+    assert v1.project(v2) == Matrix(1, 3, [R(39)/25, R(52)/25, R(13)/5])
+    assert Matrix.zeros(1, 2) == Matrix(1, 2, [0, 0])
+    assert ones(1, 2) == Matrix(1, 2, [1, 1])
+    assert v1.copy() == v1
+    # cofactor
+    assert eye(3) == eye(3).cofactor_matrix()
+    test = Matrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
+    assert test.cofactor_matrix() == \
+        Matrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
+    test = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    assert test.cofactor_matrix() == \
+        Matrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
+
+def test_cofactor_and_minors():
+    x = Symbol('x')
+    e = Matrix(4, 4,
+        [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
+
+    m = Matrix([
+        [ x,  1,  3],
+        [ 2,  9, 11],
+        [12, 13, 14]])
+    cm = Matrix([
+        [ 4,         76,       -122,        48],
+        [-8, -14*x - 68, 17*x + 142, -4*x - 72],
+        [ 4,   14*x - 8,  -21*x + 4,       8*x],
+        [ 0,  -4*x + 24,   8*x - 48, -4*x + 24]])
+    sub = Matrix([
+            [x, 1,  2],
+            [4, 5,  6],
+            [2, 9, 10]])
+
+    assert e.minor_submatrix(1, 2) == m
+    assert e.minor_submatrix(-1, -1) == sub
+    assert e.minor(1, 2) == -17*x - 142
+    assert e.cofactor(1, 2) == 17*x + 142
+    assert e.cofactor_matrix() == cm
+    assert e.cofactor_matrix(method="bareiss") == cm
+    assert e.cofactor_matrix(method="berkowitz") == cm
+
+    raises(ValueError, lambda: e.cofactor(4, 5))
+    raises(ValueError, lambda: e.minor(4, 5))
+    raises(ValueError, lambda: e.minor_submatrix(4, 5))
+
+    a = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
+    assert a.minor_submatrix(0, 0) == Matrix([[5, 6]])
+
+    raises(ValueError, lambda:
+        Matrix(0, 0, []).minor_submatrix(0, 0))
+    raises(NonSquareMatrixError, lambda: a.cofactor(0, 0))
+    raises(NonSquareMatrixError, lambda: a.minor(0, 0))
+    raises(NonSquareMatrixError, lambda: a.cofactor_matrix())
+
+def test_charpoly():
+    x, y = Symbol('x'), Symbol('y')
+    z, t = Symbol('z'), Symbol('t')
+
+    from sympy.abc import a,b,c
+
+    m = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
+
+    assert eye_Determinant(3).charpoly(x) == Poly((x - 1)**3, x)
+    assert eye_Determinant(3).charpoly(y) == Poly((y - 1)**3, y)
+    assert m.charpoly() == Poly(x**3 - 15*x**2 - 18*x, x)
+    raises(NonSquareMatrixError, lambda: Matrix([[1], [2]]).charpoly())
+    n = Matrix(4, 4, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
+    assert n.charpoly() == Poly(x**4, x)
+
+    n = Matrix(4, 4, [45, 0, 0, 0, 0, 23, 0, 0, 0, 0, 87, 0, 0, 0, 0, 12])
+    assert n.charpoly() == Poly(x**4 - 167*x**3 + 8811*x**2 - 173457*x + 1080540, x)
+
+    n = Matrix(3, 3, [x, 0, 0, a, y, 0, b, c, z])
+    assert n.charpoly() == Poly(t**3 - (x+y+z)*t**2 + t*(x*y+y*z+x*z) - x*y*z, t)
diff --git a/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_eigen.py b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_eigen.py
new file mode 100644
index 0000000000000000000000000000000000000000..75ec80bcdd55e384a1fec19d601d4aaf70b9420a
--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_eigen.py
@@ -0,0 +1,707 @@
+from sympy.core.evalf import N
+from sympy.core.numbers import (Float, I, Rational)
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.matrices import eye, Matrix
+from sympy.core.singleton import S
+from sympy.testing.pytest import raises, XFAIL
+from sympy.matrices.matrices import NonSquareMatrixError, MatrixError
+from sympy.matrices.expressions.fourier import DFT
+from sympy.simplify.simplify import simplify
+from sympy.matrices.immutable import ImmutableMatrix
+from sympy.testing.pytest import slow
+from sympy.testing.matrices import allclose
+
+
+def test_eigen():
+    R = Rational
+    M = Matrix.eye(3)
+    assert M.eigenvals(multiple=False) == {S.One: 3}
+    assert M.eigenvals(multiple=True) == [1, 1, 1]
+
+    assert M.eigenvects() == (
+        [(1, 3, [Matrix([1, 0, 0]),
+                 Matrix([0, 1, 0]),
+                 Matrix([0, 0, 1])])])
+
+    assert M.left_eigenvects() == (
+        [(1, 3, [Matrix([[1, 0, 0]]),
+                 Matrix([[0, 1, 0]]),
+                 Matrix([[0, 0, 1]])])])
+
+    M = Matrix([[0, 1, 1],
+                [1, 0, 0],
+                [1, 1, 1]])
+
+    assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
+
+    assert M.eigenvects() == (
+        [
+            (-1, 1, [Matrix([-1, 1, 0])]),
+            ( 0, 1, [Matrix([0, -1, 1])]),
+            ( 2, 1, [Matrix([R(2, 3), R(1, 3), 1])])
+        ])
+
+    assert M.left_eigenvects() == (
+        [
+            (-1, 1, [Matrix([[-2, 1, 1]])]),
+            (0, 1, [Matrix([[-1, -1, 1]])]),
+            (2, 1, [Matrix([[1, 1, 1]])])
+        ])
+
+    a = Symbol('a')
+    M = Matrix([[a, 0],
+                [0, 1]])
+
+    assert M.eigenvals() == {a: 1, S.One: 1}
+
+    M = Matrix([[1, -1],
+                [1,  3]])
+    assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
+    assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
+
+    M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    a = R(15, 2)
+    b = 3*33**R(1, 2)
+    c = R(13, 2)
+    d = (R(33, 8) + 3*b/8)
+    e = (R(33, 8) - 3*b/8)
+
+    def NS(e, n):
+        return str(N(e, n))
+    r = [
+        (a - b/2, 1, [Matrix([(12 + 24/(c - b/2))/((c - b/2)*e) + 3/(c - b/2),
+                              (6 + 12/(c - b/2))/e, 1])]),
+        (      0, 1, [Matrix([1, -2, 1])]),
+        (a + b/2, 1, [Matrix([(12 + 24/(c + b/2))/((c + b/2)*d) + 3/(c + b/2),
+                              (6 + 12/(c + b/2))/d, 1])]),
+    ]
+    r1 = [(NS(r[i][0], 2), NS(r[i][1], 2),
+        [NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
+    r = M.eigenvects()
+    r2 = [(NS(r[i][0], 2), NS(r[i][1], 2),
+        [NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
+    assert sorted(r1) == sorted(r2)
+
+    eps = Symbol('eps', real=True)
+
+    M = Matrix([[abs(eps), I*eps    ],
+                [-I*eps,   abs(eps) ]])
+
+    assert M.eigenvects() == (
+        [
+            ( 0, 1, [Matrix([[-I*eps/abs(eps)], [1]])]),
+            ( 2*abs(eps), 1, [ Matrix([[I*eps/abs(eps)], [1]]) ] ),
+        ])
+
+    assert M.left_eigenvects() == (
+        [
+            (0, 1, [Matrix([[I*eps/Abs(eps), 1]])]),
+            (2*Abs(eps), 1, [Matrix([[-I*eps/Abs(eps), 1]])])
+        ])
+
+    M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
+    M._eigenvects = M.eigenvects(simplify=False)
+    assert max(i.q for i in M._eigenvects[0][2][0]) > 1
+    M._eigenvects = M.eigenvects(simplify=True)
+    assert max(i.q for i in M._eigenvects[0][2][0]) == 1
+
+    M = Matrix([[Rational(1, 4), 1], [1, 1]])
+    assert M.eigenvects() == [
+        (Rational(5, 8) - sqrt(73)/8, 1, [Matrix([[-sqrt(73)/8 - Rational(3, 8)], [1]])]),
+        (Rational(5, 8) + sqrt(73)/8, 1, [Matrix([[Rational(-3, 8) + sqrt(73)/8], [1]])])]
+
+    # issue 10719
+    assert Matrix([]).eigenvals() == {}
+    assert Matrix([]).eigenvals(multiple=True) == []
+    assert Matrix([]).eigenvects() == []
+
+    # issue 15119
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(
+               error_when_incomplete = False))
+    raises(NonSquareMatrixError,
+           lambda: Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(
+               error_when_incomplete = False))
+
+    m = Matrix([[1, 2], [3, 4]])
+    assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
+    assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
+    assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
+    assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
+
+
+@slow
+def test_eigen_slow():
+    # issue 15125
+    from sympy.core.function import count_ops
+    q = Symbol("q", positive = True)
+    m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
+    assert count_ops(m.eigenvals(simplify=False)) > \
+        count_ops(m.eigenvals(simplify=True))
+    assert count_ops(m.eigenvals(simplify=lambda x: x)) > \
+        count_ops(m.eigenvals(simplify=True))
+
+
+def test_float_eigenvals():
+    m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
+    evals = [
+        Rational(5, 4) - sqrt(385)/20,
+        sqrt(385)/20 + Rational(5, 4),
+        S.Zero]
+
+    n_evals = m.eigenvals(rational=True, multiple=True)
+    n_evals = sorted(n_evals)
+    s_evals = [x.evalf() for x in evals]
+    s_evals = sorted(s_evals)
+
+    for x, y in zip(n_evals, s_evals):
+        assert abs(x-y) < 10**-9
+
+
+@XFAIL
+def test_eigen_vects():
+    m = Matrix(2, 2, [1, 0, 0, I])
+    raises(NotImplementedError, lambda: m.is_diagonalizable(True))
+    # !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
+    # see issue 5292
+    assert not m.is_diagonalizable(True)
+    raises(MatrixError, lambda: m.diagonalize(True))
+    (P, D) = m.diagonalize(True)
+
+def test_issue_8240():
+    # Eigenvalues of large triangular matrices
+    x, y = symbols('x y')
+    n = 200
+
+    diagonal_variables = [Symbol('x%s' % i) for i in range(n)]
+    M = [[0 for i in range(n)] for j in range(n)]
+    for i in range(n):
+        M[i][i] = diagonal_variables[i]
+    M = Matrix(M)
+
+    eigenvals = M.eigenvals()
+    assert len(eigenvals) == n
+    for i in range(n):
+        assert eigenvals[diagonal_variables[i]] == 1
+
+    eigenvals = M.eigenvals(multiple=True)
+    assert set(eigenvals) == set(diagonal_variables)
+
+    # with multiplicity
+    M = Matrix([[x, 0, 0], [1, y, 0], [2, 3, x]])
+    eigenvals = M.eigenvals()
+    assert eigenvals == {x: 2, y: 1}
+
+    eigenvals = M.eigenvals(multiple=True)
+    assert len(eigenvals) == 3
+    assert eigenvals.count(x) == 2
+    assert eigenvals.count(y) == 1
+
+
+def test_eigenvals():
+    M = Matrix([[0, 1, 1],
+                [1, 0, 0],
+                [1, 1, 1]])
+    assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
+
+    m = Matrix([
+        [3,  0,  0, 0, -3],
+        [0, -3, -3, 0,  3],
+        [0,  3,  0, 3,  0],
+        [0,  0,  3, 0,  3],
+        [3,  0,  0, 3,  0]])
+
+    # XXX Used dry-run test because arbitrary symbol that appears in
+    # CRootOf may not be unique.
+    assert m.eigenvals()
+
+
+def test_eigenvects():
+    M = Matrix([[0, 1, 1],
+                [1, 0, 0],
+                [1, 1, 1]])
+    vecs = M.eigenvects()
+    for val, mult, vec_list in vecs:
+        assert len(vec_list) == 1
+        assert M*vec_list[0] == val*vec_list[0]
+
+
+def test_left_eigenvects():
+    M = Matrix([[0, 1, 1],
+                [1, 0, 0],
+                [1, 1, 1]])
+    vecs = M.left_eigenvects()
+    for val, mult, vec_list in vecs:
+        assert len(vec_list) == 1
+        assert vec_list[0]*M == val*vec_list[0]
+
+
+@slow
+def test_bidiagonalize():
+    M = Matrix([[1, 0, 0],
+                [0, 1, 0],
+                [0, 0, 1]])
+    assert M.bidiagonalize() == M
+    assert M.bidiagonalize(upper=False) == M
+    assert M.bidiagonalize() == M
+    assert M.bidiagonal_decomposition() == (M, M, M)
+    assert M.bidiagonal_decomposition(upper=False) == (M, M, M)
+    assert M.bidiagonalize() == M
+
+    import random
+    #Real Tests
+    for real_test in range(2):
+        test_values = []
+        row = 2
+        col = 2
+        for _ in range(row * col):
+            value = random.randint(-1000000000, 1000000000)
+            test_values = test_values + [value]
+        # L     -> Lower Bidiagonalization
+        # M     -> Mutable Matrix
+        # N     -> Immutable Matrix
+        # 0     -> Bidiagonalized form
+        # 1,2,3 -> Bidiagonal_decomposition matrices
+        # 4     -> Product of 1 2 3
+        M = Matrix(row, col, test_values)
+        N = ImmutableMatrix(M)
+
+        N1, N2, N3 = N.bidiagonal_decomposition()
+        M1, M2, M3 = M.bidiagonal_decomposition()
+        M0 = M.bidiagonalize()
+        N0 = N.bidiagonalize()
+
+        N4 = N1 * N2 * N3
+        M4 = M1 * M2 * M3
+
+        N2.simplify()
+        N4.simplify()
+        N0.simplify()
+
+        M0.simplify()
+        M2.simplify()
+        M4.simplify()
+
+        LM0 = M.bidiagonalize(upper=False)
+        LM1, LM2, LM3 = M.bidiagonal_decomposition(upper=False)
+        LN0 = N.bidiagonalize(upper=False)
+        LN1, LN2, LN3 = N.bidiagonal_decomposition(upper=False)
+
+        LN4 = LN1 * LN2 * LN3
+        LM4 = LM1 * LM2 * LM3
+
+        LN2.simplify()
+        LN4.simplify()
+        LN0.simplify()
+
+        LM0.simplify()
+        LM2.simplify()
+        LM4.simplify()
+
+        assert M == M4
+        assert M2 == M0
+        assert N == N4
+        assert N2 == N0
+        assert M == LM4
+        assert LM2 == LM0
+        assert N == LN4
+        assert LN2 == LN0
+
+    #Complex Tests
+    for complex_test in range(2):
+        test_values = []
+        size = 2
+        for _ in range(size * size):
+            real = random.randint(-1000000000, 1000000000)
+            comp = random.randint(-1000000000, 1000000000)
+            value = real + comp * I
+            test_values = test_values + [value]
+        M = Matrix(size, size, test_values)
+        N = ImmutableMatrix(M)
+        # L     -> Lower Bidiagonalization
+        # M     -> Mutable Matrix
+        # N     -> Immutable Matrix
+        # 0     -> Bidiagonalized form
+        # 1,2,3 -> Bidiagonal_decomposition matrices
+        # 4     -> Product of 1 2 3
+        N1, N2, N3 = N.bidiagonal_decomposition()
+        M1, M2, M3 = M.bidiagonal_decomposition()
+        M0 = M.bidiagonalize()
+        N0 = N.bidiagonalize()
+
+        N4 = N1 * N2 * N3
+        M4 = M1 * M2 * M3
+
+        N2.simplify()
+        N4.simplify()
+        N0.simplify()
+
+        M0.simplify()
+        M2.simplify()
+        M4.simplify()
+
+        LM0 = M.bidiagonalize(upper=False)
+        LM1, LM2, LM3 = M.bidiagonal_decomposition(upper=False)
+        LN0 = N.bidiagonalize(upper=False)
+        LN1, LN2, LN3 = N.bidiagonal_decomposition(upper=False)
+
+        LN4 = LN1 * LN2 * LN3
+        LM4 = LM1 * LM2 * LM3
+
+        LN2.simplify()
+        LN4.simplify()
+        LN0.simplify()
+
+        LM0.simplify()
+        LM2.simplify()
+        LM4.simplify()
+
+        assert M == M4
+        assert M2 == M0
+        assert N == N4
+        assert N2 == N0
+        assert M == LM4
+        assert LM2 == LM0
+        assert N == LN4
+        assert LN2 == LN0
+
+    M = Matrix(18, 8, range(1, 145))
+    M = M.applyfunc(lambda i: Float(i))
+    assert M.bidiagonal_decomposition()[1] == M.bidiagonalize()
+    assert M.bidiagonal_decomposition(upper=False)[1] == M.bidiagonalize(upper=False)
+    a, b, c = M.bidiagonal_decomposition()
+    diff = a * b * c - M
+    assert abs(max(diff)) < 10**-12
+
+
+def test_diagonalize():
+    m = Matrix(2, 2, [0, -1, 1, 0])
+    raises(MatrixError, lambda: m.diagonalize(reals_only=True))
+    P, D = m.diagonalize()
+    assert D.is_diagonal()
+    assert D == Matrix([
+                 [-I, 0],
+                 [ 0, I]])
+
+    # make sure we use floats out if floats are passed in
+    m = Matrix(2, 2, [0, .5, .5, 0])
+    P, D = m.diagonalize()
+    assert all(isinstance(e, Float) for e in D.values())
+    assert all(isinstance(e, Float) for e in P.values())
+
+    _, D2 = m.diagonalize(reals_only=True)
+    assert D == D2
+
+    m = Matrix(
+        [[0, 1, 0, 0], [1, 0, 0, 0.002], [0.002, 0, 0, 1], [0, 0, 1, 0]])
+    P, D = m.diagonalize()
+    assert allclose(P*D, m*P)
+
+
+def test_is_diagonalizable():
+    a, b, c = symbols('a b c')
+    m = Matrix(2, 2, [a, c, c, b])
+    assert m.is_symmetric()
+    assert m.is_diagonalizable()
+    assert not Matrix(2, 2, [1, 1, 0, 1]).is_diagonalizable()
+
+    m = Matrix(2, 2, [0, -1, 1, 0])
+    assert m.is_diagonalizable()
+    assert not m.is_diagonalizable(reals_only=True)
+
+
+def test_jordan_form():
+    m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
+    raises(NonSquareMatrixError, lambda: m.jordan_form())
+
+    # the next two tests test the cases where the old
+    # algorithm failed due to the fact that the block structure can
+    # *NOT* be determined  from algebraic and geometric multiplicity alone
+    # This can be seen most easily when one lets compute the J.c.f. of a matrix that
+    # is in J.c.f already.
+    m = Matrix(4, 4, [2, 1, 0, 0,
+                    0, 2, 1, 0,
+                    0, 0, 2, 0,
+                    0, 0, 0, 2
+    ])
+    P, J = m.jordan_form()
+    assert m == J
+
+    m = Matrix(4, 4, [2, 1, 0, 0,
+                    0, 2, 0, 0,
+                    0, 0, 2, 1,
+                    0, 0, 0, 2
+    ])
+    P, J = m.jordan_form()
+    assert m == J
+
+    A = Matrix([[ 2,  4,  1,  0],
+                [-4,  2,  0,  1],
+                [ 0,  0,  2,  4],
+                [ 0,  0, -4,  2]])
+    P, J = A.jordan_form()
+    assert simplify(P*J*P.inv()) == A
+
+    assert Matrix(1, 1, [1]).jordan_form() == (Matrix([1]), Matrix([1]))
+    assert Matrix(1, 1, [1]).jordan_form(calc_transform=False) == Matrix([1])
+
+    # If we have eigenvalues in CRootOf form, raise errors
+    m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
+    raises(MatrixError, lambda: m.jordan_form())
+
+    # make sure that if the input has floats, the output does too
+    m = Matrix([
+        [                0.6875, 0.125 + 0.1875*sqrt(3)],
+        [0.125 + 0.1875*sqrt(3),                 0.3125]])
+    P, J = m.jordan_form()
+    assert all(isinstance(x, Float) or x == 0 for x in P)
+    assert all(isinstance(x, Float) or x == 0 for x in J)
+
+
+def test_singular_values():
+    x = Symbol('x', real=True)
+
+    A = Matrix([[0, 1*I], [2, 0]])
+    # if singular values can be sorted, they should be in decreasing order
+    assert A.singular_values() == [2, 1]
+
+    A = eye(3)
+    A[1, 1] = x
+    A[2, 2] = 5
+    vals = A.singular_values()
+    # since Abs(x) cannot be sorted, test set equality
+    assert set(vals) == {5, 1, Abs(x)}
+
+    A = Matrix([[sin(x), cos(x)], [-cos(x), sin(x)]])
+    vals = [sv.trigsimp() for sv in A.singular_values()]
+    assert vals == [S.One, S.One]
+
+    A = Matrix([
+        [2, 4],
+        [1, 3],
+        [0, 0],
+        [0, 0]
+        ])
+    assert A.singular_values() == \
+        [sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221))]
+    assert A.T.singular_values() == \
+        [sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221)), 0, 0]
+
+def test___eq__():
+    assert (Matrix(
+        [[0, 1, 1],
+        [1, 0, 0],
+        [1, 1, 1]]) == {}) is False
+
+
+def test_definite():
+    # Examples from Gilbert Strang, "Introduction to Linear Algebra"
+    # Positive definite matrices
+    m = Matrix([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    m = Matrix([[5, 4], [4, 5]])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    # Positive semidefinite matrices
+    m = Matrix([[2, -1, -1], [-1, 2, -1], [-1, -1, 2]])
+    assert m.is_positive_definite == False
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    m = Matrix([[1, 2], [2, 4]])
+    assert m.is_positive_definite == False
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    # Examples from Mathematica documentation
+    # Non-hermitian positive definite matrices
+    m = Matrix([[2, 3], [4, 8]])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    # Hermetian matrices
+    m = Matrix([[1, 2*I], [-I, 4]])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    # Symbolic matrices examples
+    a = Symbol('a', positive=True)
+    b = Symbol('b', negative=True)
+    m = Matrix([[a, 0, 0], [0, a, 0], [0, 0, a]])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == False
+
+    m = Matrix([[b, 0, 0], [0, b, 0], [0, 0, b]])
+    assert m.is_positive_definite == False
+    assert m.is_positive_semidefinite == False
+    assert m.is_negative_definite == True
+    assert m.is_negative_semidefinite == True
+    assert m.is_indefinite == False
+
+    m = Matrix([[a, 0], [0, b]])
+    assert m.is_positive_definite == False
+    assert m.is_positive_semidefinite == False
+    assert m.is_negative_definite == False
+    assert m.is_negative_semidefinite == False
+    assert m.is_indefinite == True
+
+    m = Matrix([
+        [0.0228202735623867, 0.00518748979085398,
+         -0.0743036351048907, -0.00709135324903921],
+        [0.00518748979085398, 0.0349045359786350,
+         0.0830317991056637, 0.00233147902806909],
+        [-0.0743036351048907, 0.0830317991056637,
+         1.15859676366277, 0.340359081555988],
+        [-0.00709135324903921, 0.00233147902806909,
+         0.340359081555988, 0.928147644848199]
+    ])
+    assert m.is_positive_definite == True
+    assert m.is_positive_semidefinite == True
+    assert m.is_indefinite == False
+
+    # test for issue 19547: https://github.com/sympy/sympy/issues/19547
+    m = Matrix([
+        [0, 0, 0],
+        [0, 1, 2],
+        [0, 2, 1]
+    ])
+    assert not m.is_positive_definite
+    assert not m.is_positive_semidefinite
+
+
+def test_positive_semidefinite_cholesky():
+    from sympy.matrices.eigen import _is_positive_semidefinite_cholesky
+
+    m = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
+    assert _is_positive_semidefinite_cholesky(m) == True
+    m = Matrix([[0, 0, 0], [0, 5, -10*I], [0, 10*I, 5]])
+    assert _is_positive_semidefinite_cholesky(m) == False
+    m = Matrix([[1, 0, 0], [0, 0, 0], [0, 0, -1]])
+    assert _is_positive_semidefinite_cholesky(m) == False
+    m = Matrix([[0, 1], [1, 0]])
+    assert _is_positive_semidefinite_cholesky(m) == False
+
+    # https://www.value-at-risk.net/cholesky-factorization/
+    m = Matrix([[4, -2, -6], [-2, 10, 9], [-6, 9, 14]])
+    assert _is_positive_semidefinite_cholesky(m) == True
+    m = Matrix([[9, -3, 3], [-3, 2, 1], [3, 1, 6]])
+    assert _is_positive_semidefinite_cholesky(m) == True
+    m = Matrix([[4, -2, 2], [-2, 1, -1], [2, -1, 5]])
+    assert _is_positive_semidefinite_cholesky(m) == True
+    m = Matrix([[1, 2, -1], [2, 5, 1], [-1, 1, 9]])
+    assert _is_positive_semidefinite_cholesky(m) == False
+
+
+def test_issue_20582():
+    A = Matrix([
+        [5, -5, -3, 2, -7],
+        [-2, -5, 0, 2, 1],
+        [-2, -7, -5, -2, -6],
+        [7, 10, 3, 9, -2],
+        [4, -10, 3, -8, -4]
+    ])
+    # XXX Used dry-run test because arbitrary symbol that appears in
+    # CRootOf may not be unique.
+    assert A.eigenvects()
+
+def test_issue_19210():
+    t = Symbol('t')
+    H = Matrix([[3, 0, 0, 0], [0, 1 , 2, 0], [0, 2, 2, 0], [0, 0, 0, 4]])
+    A = (-I * H * t).jordan_form()
+    assert A == (Matrix([
+                    [0, 1,                  0,                0],
+                    [0, 0, -4/(-1 + sqrt(17)), 4/(1 + sqrt(17))],
+                    [0, 0,                  1,                1],
+                    [1, 0,                  0,                0]]), Matrix([
+                    [-4*I*t,      0,                         0,                         0],
+                    [     0, -3*I*t,                         0,                         0],
+                    [     0,      0, t*(-3*I/2 + sqrt(17)*I/2),                         0],
+                    [     0,      0,                         0, t*(-sqrt(17)*I/2 - 3*I/2)]]))
+
+
+def test_issue_20275():
+    # XXX We use complex expansions because complex exponentials are not
+    # recognized by polys.domains
+    A = DFT(3).as_explicit().expand(complex=True)
+    eigenvects = A.eigenvects()
+    assert eigenvects[0] == (
+        -1, 1,
+        [Matrix([[1 - sqrt(3)], [1], [1]])]
+    )
+    assert eigenvects[1] == (
+        1, 1,
+        [Matrix([[1 + sqrt(3)], [1], [1]])]
+    )
+    assert eigenvects[2] == (
+        -I, 1,
+        [Matrix([[0], [-1], [1]])]
+    )
+
+    A = DFT(4).as_explicit().expand(complex=True)
+    eigenvects = A.eigenvects()
+    assert eigenvects[0] == (
+        -1, 1,
+        [Matrix([[-1], [1], [1], [1]])]
+    )
+    assert eigenvects[1] == (
+        1, 2,
+        [Matrix([[1], [0], [1], [0]]), Matrix([[2], [1], [0], [1]])]
+    )
+    assert eigenvects[2] == (
+        -I, 1,
+        [Matrix([[0], [-1], [0], [1]])]
+    )
+
+    # XXX We skip test for some parts of eigenvectors which are very
+    # complicated and fragile under expression tree changes
+    A = DFT(5).as_explicit().expand(complex=True)
+    eigenvects = A.eigenvects()
+    assert eigenvects[0] == (
+        -1, 1,
+        [Matrix([[1 - sqrt(5)], [1], [1], [1], [1]])]
+    )
+    assert eigenvects[1] == (
+        1, 2,
+        [Matrix([[S(1)/2 + sqrt(5)/2], [0], [1], [1], [0]]),
+         Matrix([[S(1)/2 + sqrt(5)/2], [1], [0], [0], [1]])]
+    )
+
+
+def test_issue_20752():
+    b = symbols('b', nonzero=True)
+    m = Matrix([[0, 0, 0], [0, b, 0], [0, 0, b]])
+    assert m.is_positive_semidefinite is None
diff --git a/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_graph.py b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_graph.py
new file mode 100644
index 0000000000000000000000000000000000000000..0bf3c819a9477387f53560a034d7949fd76a654f
--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_graph.py
@@ -0,0 +1,108 @@
+from sympy.combinatorics import Permutation
+from sympy.core.symbol import symbols
+from sympy.matrices import Matrix
+from sympy.matrices.expressions import (
+    PermutationMatrix, BlockDiagMatrix, BlockMatrix)
+
+
+def test_connected_components():
+    a, b, c, d, e, f, g, h, i, j, k, l, m = symbols('a:m')
+
+    M = Matrix([
+        [a, 0, 0, 0, b, 0, 0, 0, 0, 0, c, 0, 0],
+        [0, d, 0, 0, 0, e, 0, 0, 0, 0, 0, f, 0],
+        [0, 0, g, 0, 0, 0, h, 0, 0, 0, 0, 0, i],
+        [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+        [m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
+        [0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
+        [0, 0, m, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
+        [j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0, 0],
+        [0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l, 0],
+        [0, 0, j, 0, 0, 0, k, 0, 0, 1, 0, 0, l],
+        [0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0, 0],
+        [0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1, 0],
+        [0, 0, 0, 0, 0, 0, d, 0, 0, 0, 0, 0, 1]])
+    cc = M.connected_components()
+    assert cc == [[0, 4, 7, 10], [1, 5, 8, 11], [2, 6, 9, 12], [3]]
+
+    P, B = M.connected_components_decomposition()
+    p = Permutation([0, 4, 7, 10, 1, 5, 8, 11, 2, 6, 9, 12, 3])
+    assert P == PermutationMatrix(p)
+
+    B0 = Matrix([
+        [a, b, 0, c],
+        [m, 1, 0, 0],
+        [j, k, 1, l],
+        [0, d, 0, 1]])
+    B1 = Matrix([
+        [d, e, 0, f],
+        [m, 1, 0, 0],
+        [j, k, 1, l],
+        [0, d, 0, 1]])
+    B2 = Matrix([
+        [g, h, 0, i],
+        [m, 1, 0, 0],
+        [j, k, 1, l],
+        [0, d, 0, 1]])
+    B3 = Matrix([[1]])
+    assert B == BlockDiagMatrix(B0, B1, B2, B3)
+
+
+def test_strongly_connected_components():
+    M = Matrix([
+        [11, 14, 10, 0, 15, 0],
+        [0, 44, 0, 0, 45, 0],
+        [1, 4, 0, 0, 5, 0],
+        [0, 0, 0, 22, 0, 23],
+        [0, 54, 0, 0, 55, 0],
+        [0, 0, 0, 32, 0, 33]])
+    scc = M.strongly_connected_components()
+    assert scc == [[1, 4], [0, 2], [3, 5]]
+
+    P, B = M.strongly_connected_components_decomposition()
+    p = Permutation([1, 4, 0, 2, 3, 5])
+    assert P == PermutationMatrix(p)
+    assert B == BlockMatrix([
+        [
+            Matrix([[44, 45], [54, 55]]),
+            Matrix.zeros(2, 2),
+            Matrix.zeros(2, 2)
+        ],
+        [
+            Matrix([[14, 15], [4, 5]]),
+            Matrix([[11, 10], [1, 0]]),
+            Matrix.zeros(2, 2)
+        ],
+        [
+            Matrix.zeros(2, 2),
+            Matrix.zeros(2, 2),
+            Matrix([[22, 23], [32, 33]])
+        ]
+    ])
+    P = P.as_explicit()
+    B = B.as_explicit()
+    assert P.T * B * P == M
+
+    P, B = M.strongly_connected_components_decomposition(lower=False)
+    p = Permutation([3, 5, 0, 2, 1, 4])
+    assert P == PermutationMatrix(p)
+    assert B == BlockMatrix([
+        [
+            Matrix([[22, 23], [32, 33]]),
+            Matrix.zeros(2, 2),
+            Matrix.zeros(2, 2)
+        ],
+        [
+            Matrix.zeros(2, 2),
+            Matrix([[11, 10], [1, 0]]),
+            Matrix([[14, 15], [4, 5]])
+        ],
+        [
+            Matrix.zeros(2, 2),
+            Matrix.zeros(2, 2),
+            Matrix([[44, 45], [54, 55]])
+        ]
+    ])
+    P = P.as_explicit()
+    B = B.as_explicit()
+    assert P.T * B * P == M
diff --git a/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_interactions.py b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_interactions.py
new file mode 100644
index 0000000000000000000000000000000000000000..98da16bc282da923e6aa9aeb45c78735615951b0
--- /dev/null
+++ b/llmeval-env/lib/python3.10/site-packages/sympy/matrices/tests/test_interactions.py
@@ -0,0 +1,77 @@
+"""
+We have a few different kind of Matrices
+Matrix, ImmutableMatrix, MatrixExpr
+
+Here we test the extent to which they cooperate
+"""
+
+from sympy.core.symbol import symbols
+from sympy.matrices import (Matrix, MatrixSymbol, eye, Identity,
+        ImmutableMatrix)
+from sympy.matrices.expressions import MatrixExpr, MatAdd
+from sympy.matrices.common import classof
+from sympy.testing.pytest import raises
+
+SM = MatrixSymbol('X', 3, 3)
+SV = MatrixSymbol('v', 3, 1)
+MM = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+IM = ImmutableMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+meye = eye(3)
+imeye = ImmutableMatrix(eye(3))
+ideye = Identity(3)
+a, b, c = symbols('a,b,c')
+
+
+def test_IM_MM():
+    assert isinstance(MM + IM, ImmutableMatrix)
+    assert isinstance(IM + MM, ImmutableMatrix)
+    assert isinstance(2*IM + MM, ImmutableMatrix)
+    assert MM.equals(IM)
+
+
+def test_ME_MM():
+    assert isinstance(Identity(3) + MM, MatrixExpr)
+    assert isinstance(SM + MM, MatAdd)
+    assert isinstance(MM + SM, MatAdd)
+    assert (Identity(3) + MM)[1, 1] == 6
+
+
+def test_equality():
+    a, b, c = Identity(3), eye(3), ImmutableMatrix(eye(3))
+    for x in [a, b, c]:
+        for y in [a, b, c]:
+            assert x.equals(y)
+
+
+def test_matrix_symbol_MM():
+    X = MatrixSymbol('X', 3, 3)
+    Y = eye(3) + X
+    assert Y[1, 1] == 1 + X[1, 1]
+
+
+def test_matrix_symbol_vector_matrix_multiplication():
+    A = MM * SV
+    B = IM * SV
+    assert A == B
+    C = (SV.T * MM.T).T
+    assert B == C
+    D = (SV.T * IM.T).T
+    assert C == D
+
+
+def test_indexing_interactions():
+    assert (a * IM)[1, 1] == 5*a
+    assert (SM + IM)[1, 1] == SM[1, 1] + IM[1, 1]
+    assert (SM * IM)[1, 1] == SM[1, 0]*IM[0, 1] + SM[1, 1]*IM[1, 1] + \
+        SM[1, 2]*IM[2, 1]
+
+
+def test_classof():
+    A = Matrix(3, 3, range(9))
+    B = ImmutableMatrix(3, 3, range(9))
+    C = MatrixSymbol('C', 3, 3)
+    assert classof(A, A) == Matrix
+    assert classof(B, B) == ImmutableMatrix
+    assert classof(A, B) == ImmutableMatrix
+    assert classof(B, A) == ImmutableMatrix
+    raises(TypeError, lambda: classof(A, C))
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