This is about exercise 1207 from the book "Problems and Solutions in Mathematics", 2nd edition, by Ta-Tsien.
Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over the finite field $F_p$. Let $G = GL_{F_p}(V)$. Prove that $T \in G$ is semisimple if and only if $T^{p^m-1}=1$ for some positive integer $m$.
The solution of this question starts with that:
Let $T \in G$ and $f(\lambda)$ be the minimal polynomial of $T$ over $F_p$. Then $\lambda \nmid f(\lambda)$.
I fail to see how they deduce this assertion about $\lambda$. For example, if $T$ is the null matrix, does the assertion still holds ?
Suppose $\lambda \mid f(\lambda)$, then 0 is a root of the minimal polynomial $f(\lambda)$. The set of roots of the minimal polynomial is equal to the set of eigenvalues of $T$. Therefore at least one of the eigenvalues of $T$ is equal to 0. Contradiction with $T \in GL(V)$ ensues.