## If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent

## Problem 21

Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.

Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.