Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
EXAMPLES:
– "hessenberg" - Use the Hessenberg form of the matrix
sage: A = MatrixSpace(Integers(8),3)([1,7,3, 1,1,1, 3,4,5])
sage: A.determinant()
6
sage: A.determinant() is A.determinant()
True
sage: A[0,0] = 10
sage: A.determinant()
7
We compute the determinant of the arbitrary 3x3 matrix:
sage: R = PolynomialRing(QQ,9,’x’)
sage: A = matrix(R,3,R.gens())
sage: A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: A.determinant()
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8
We create a matrix over Z[x, y] and compute its determinant.
sage: R. = PolynomialRing(IntegerRing(),2)
sage: A = MatrixSpace(R,2)([x, y, x**2, y**2])
sage: A.determinant()
-x^2*y + x*y^2
TESTS:
sage: A = matrix(5, 5, [next_prime(i^2) for i in range(25)])
sage: B = MatrixSpace(ZZ[’x’], 5, 5)(A)
sage: A.det() - B.det()
0
We verify that trac ticket #5569 is resolved (otherwise the following would hang for hours):
sage: d = random_matrix(GF(next_prime(10^20)),50).det()
sage: d = random_matrix(Integers(10^50),50).det()
We verify that trac 7704 is resolved:
sage: matrix(ZZ, {(0,0):1,(1,1):2,(2,2):3,(3,3):4}).det()
24
sage: matrix(QQ, {(0,0):1,(1,1):2,(2,2):3,(3,3):4}).det()
24
We verify that trac 10063 is resolved:
sage: A = GF(2)[’x,y,z’]
sage: A.inject_variables()
Defining x, y, z
sage: R = A.quotient(x^2 + 1).quotient(y^2 + 1).quotient(z^2 + 1)
sage: R.inject_variables()
Defining xbarbarbar, ybarbarbar, zbarbarbar
sage: M = matrix([[1,1,1,1],[xbarbarbar,ybarbarbar,1,1],[0,1,zbarbarbar,1],[xbarbarbar,zbarb
sage: M.determinant()
xbarbarbar*ybarbarbar*zbarbarbar + xbarbarbar*ybarbarbar + xbarbarbar*zbarbarbar + ybarbarba
151