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_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]])