Let $A \in \mathrm{M}_{2}(\mathbb{Z})$ be such that $\left|A_{i j}(n)\right| \leq 50$ for all $1 \leq n \leq 10^{50}$ and all $1 \leq i, j \leq 2$, where $A_{i j}(n)$ denotes the $(i, j)$-th entry of the $2 \times 2$ matrix $A^{n}$. Then $\left|A_{i j}(n)\right| \leq 50$ for all positive integers $n$.
