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