TITLE: To be a scalar matrix or not to be?
QUESTION [7 upvotes]: What follows is a pretty (but not so easy) exercise. Is fun off-topic?

Let $A, B \in M_2(\mathbb{C})$. Show that for every $m, n \in \mathbb{N}$ $$((AB)^m-(BA)^m)((AB)^n-(BA)^n)$$ is a scalar matrix (that is, of the form $\lambda I_2$).

REPLY [7 votes]: I don't know what trick you had in mind, but it is sufficient to note that $((AB)^m - (BA)^m)$ and $((AB)^n-(BA)^n)$ are commuting matrices, each with trace $0$.