A=matrix(QQ, 3, 3, [1, 0, 0, 1, 3, 5, 8, 1, 2])
B=matrix(QQ, 3, 3, [0, 1, 2, 8, 0, 7, 9, 0, 8])
AB=A*B
I=identity_matrix(3)
Aug1=A.augment(I).rref()
Aug2=B.augment(I).rref()
Aug3=AB.augment(I).rref()
print "[I:A^(-1)]="
print Aug1
print "[I:B^(-1)]="
print Aug2
print "[I:(AB)^(-1)="
print Aug3
print AB.inverse()==B.inverse()*A.inverse()
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[I:A^(-1)]=
[ 1 0 0 1 0 0]
[ 0 1 0 38 2 -5]
[ 0 0 1 -23 -1 3]
[I:B^(-1)]=
[ 1 0 0 0 8 -7]
[ 0 1 0 1 18 -16]
[ 0 0 1 0 -9 8]
[I:(AB)^(-1)=
[ 1 0 0 465 23 -61]
[ 0 1 0 1053 52 -138]
[ 0 0 1 -526 -26 69]
True
[I:A^(-1)]=
[ 1 0 0 1 0 0]
[ 0 1 0 38 2 -5]
[ 0 0 1 -23 -1 3]
[I:B^(-1)]=
[ 1 0 0 0 8 -7]
[ 0 1 0 1 18 -16]
[ 0 0 1 0 -9 8]
[I:(AB)^(-1)=
[ 1 0 0 465 23 -61]
[ 0 1 0 1053 52 -138]
[ 0 0 1 -526 -26 69]
True
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