linear algebra - Proof that eigenvalues of matrix with zero trace are all equal to zero

I'm working on the following question:

"Suppose that $A$ is a complex square matrix such that the trace of $A^k$ is zero for every $k \in \mathbb{N}$. Show that all the eigenvalues of $A$ are zero."

I think perhaps I should use the fact that the trace of a matrix is equal to the sum of its eigenvalues? Thoughts?