An nxn matrix is invertible if there is an nxn matrix C such that CA = AC = I.
So since A^k is just A*A^(k-1) or you could say its A^(k-1)*A then A^(k-1) is that C matrix and is the inverse of A and A is invertible.
Is that correct?
Let A be an n x n matrix such that Ak = In for some positive integer k.
Prove that A is invertible.
We have studied inverses of matrices and the Invertible Matrix Theorem, but have not yet reached determinants, just to let you know that...