MathNet 2025-part-02 — 186 problems (EN–VI) Question 37

Let $P$ and $Q$ be two matrices in $M_n(\mathbb{C})$. Assume $a \in \mathbb{R} \setminus \{-1, 0, 1\}$, such that $Q^2 = (a-1)QP - (a+1)PQ$. Prove that $(PQ - QP)^n = O_n$.