Stuck on two homework problems.
1. Let A and B be n x n matricies such that AB = I. Prove that det(A) is not equal to 0 and det(B) is not equal to 0.
2. Let A and B be n x n matricies such that AB is singular. Prove that either A or B is singular.
Stuck on two homework problems.
1. Let A and B be n x n matricies such that AB = I. Prove that det(A) is not equal to 0 and det(B) is not equal to 0.
2. Let A and B be n x n matricies such that AB is singular. Prove that either A or B is singular.
1.
AB=I
det(AB)=det(I)
det(A)det(B)=det(I)
det(I)=1.
If det(A) v det(B)=0, then you get zero times another scalar, which is zero.
Hence, det(A) and det(B) ≠ 0.
QED
2.
If AB is singular, then,
det(AB)=0.
So, det(A)det(B)=0.
If neither scalars det(A) or det(B)≠0, then neither one of them could be singular, sinceall nonsingular matricies have nonzero determinants; so, det(A)det(B) wouldn't be equal to zero.
So, A is singular, or B is singular.
QED
Recall that for all scalars a b in R ab=0 iff (a=0)v(b=0)