Trace zero means matrix is nilpotent?

Question

I have to prove or disprove:

If $A$ is an $n \times n$ matrix in $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$ and the trace of any power of $A$ is $0$, then the matrix is nilpotent: $A^k = 0$ for some positive integer $k$.

I'm not sure where to go with this. But I don't think this statement is true Are there any good counterexamples?

Answer 1

Think of the $2 \times 2$ identity matrix for $p = 2$. Generalize.