## The trace of a nilpotent matrix over a field is 0

Let $N$ be an $n \times n$ matrix over a field $F$, and suppose $N^k = 0$ for some $k$. Show that $\mathsf{trace}\ N = 0$.

We showed previously that the eigenvalues of $N$ are all 0. We also showed previously that the trace of a matrix is the sum of its eigenvalues. Thus $N$ has trace 0.