Problem

Source: Romanian National Olympiad 2019 - Grade 11 - Problem 1

Tags: linear algebra, matrix



Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$ Note: $X^*$ is the adjugate matrix of $X$ (not the conjugate transpose)