Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
EXAMPLES: The following matrix is invertible over Q but not over Z.
sage: A = MatrixSpace(ZZ, 2)(range(4))
sage: A.is_invertible()
False
sage: A.matrix_over_field().is_invertible()
True
The inverse function is a constructor for matrices over the fraction field, so it can work even if A is not
invertible.
sage: ~A # inverse of A
[-3/2 1/2]
[ 1 0]
The next matrix is invertible over Z.
sage: A = MatrixSpace(IntegerRing(),2)([1,10,0,-1])
sage: A.is_invertible()
True
sage: ~A
# compute the inverse
[ 1 10]
[ 0 -1]
The following nontrivial matrix is invertible over Z[x].
sage: R. = PolynomialRing(IntegerRing())
sage: A = MatrixSpace(R,2)([1,x,0,-1])
sage: A.is_invertible()
True
sage: ~A
[ 1 x]
[ 0 -1]
is_mutable()
Return True if this matrix is mutable.
See the documentation for self.set_immutable for more details about mutability.
EXAMPLES:
sage: A = Matrix(QQ[’t’,’s’], 2, 2, range(4))
sage: A.is_mutable()
True
sage: A.set_immutable()
sage: A.is_mutable()
False
is_singular()
Returns True if self is singular.
OUTPUT:
A square matrix is singular if it has a zero determinant and this method will return True in exactly this
case. When the entries of the matrix come from a field, this is equivalent to having a nontrivial kernel, or
lacking an inverse, or having linearly dependent rows, or having linearly dependent columns.
For square matrices over a field the methods is_invertible() and is_singular() are logical
opposites. However, it is an error to apply is_singular() to a matrix that is not square, while
is_invertible() will always return False for a matrix that is not square.
EXAMPLES:
75