Suppose that $A \in M_{n \times n} (\mathbb{C})$ and $A$ is invertible. If $A^k$ is diagonalizable for some $k \in \mathbb{N}$ prove that $A$ is diagonalizable.
I have tried using the fact that the characteristic polynomial of a matrix is dividable by its minimal polynomial and if a matrix is diagonalizable, its minimal polynomial have only simple roots(roots with no repetition). But I can't figure out how to construct minimal/characteristic polynomial of $A^k$ using A or vice-versa.