MathNet 2013 Question 60

Let $m$ and $n$ be integers $m, n \ge 2$. The matrices $A_1, A_2, \dots, A_m \in \mathcal{M}_n(\mathbb{R})$, are not all nilpotent. Prove that there is an integer $k \gt 0$ such that $A_1^k + A_2^k + \dots + A_m^k \ne O_n$.

Marius Cavachi