MathNet 2019-part-05 — 184 problems (EN–VI) Question 62

Let $n$ be an odd positive integer and let $A, B \in \mathcal{M}_n(\mathbb{C})$ be two matrices with complex entries, such that $(A - B)^2 = O_n$. Prove that
$$
\det(AB - BA) = 0.
$$