Question 92
Let $n$ be a positive integer with $n \ge 2$, and let $A, B \in \mathcal{M}_n(\mathbb{C})$ be two matrices with complex entries having the property that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$, such that $C^* = AB - BA$. Prove that $(AB - BA)^2 = O_n$. (A matrix $C$ is called *idempotent* if $C^2 = C$. The matrix $C^*$ is the adjoint matrix of $C$.)