A^k matrix singularity and (A^k)^-1 = (A^-1)^k?
Let A be nonsingular. Prove That for any positive integer k , A^k is nonsingular, And (A^k)^-1 = (A^-1)^k.
- Anonymous1 decade agoFavorite Answer
Base case k=1: A^1 = A is non singular -ok-
(Note: the base case could be k=0 if you accepted A^0 = Identity matrix)
Inductive step: assume A^k non singular, prove A^(k+1) is non singular.
A^(k+1) = A^k A is a product of two non singular matrices, so its non singular -ok-. The statement used can be proved using
det(M) = 0 if and only if M is singular,
and det (M N) = det(M) det (N).
So if neither det(M) = 0 nor det(N)=0, then det(MN) is not equal 0, so MN is non singular.
The second proposal can also be demonstrated inductively. But its more intuitive to write out (A^-1)^k A^k and show that the result is the Identity matrix...
(A^-1)(A^-1)...(A^-1) A A ... A = I
As a matter of fact, the first proposition can be showed to be true this way, but is more formal if you use complete induction.