Bio Rua

My professor gave me this problem as an bonus problem. It has got me thinking for a while but I haven't come up with a solution yet. Hope I can boost this thread up a little bit.

Let A be a non singular integer square matrix with integer entries. Suppose that for every positive integer k, there is a matrix X with integer entries such that $X^{k}=A$. Prove that A must be the identity matrix

Quick note #1: Its obvious if A is an 1 by 1 matrix or A is diagonalizable.
Quick note #2: Cavachi - the owner of this problem - is pretty famous himself for proposals of difficult arithmetic problems in MAA Monthly. So if you can solve it, you can have the feeling of beating a big dude