I have a set of square matrices $A_i \in \mathbb{R}^{n \times n}$ for $i=1,\ldots,N$, such that $[A_i]_{jk} \ge 0$ for all $i$ and coordinates $j,k$.
If the largest eigenvalue of each $A_i$ is smaller than 1, is it going to be true also for $\frac{1}{N} \sum_{i=1}^N A_i?$