Question 60
a) Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A^2B = A$. Prove that
$$
(AB - BA)^2 = O_n.
$$
b) Show that, for any natural number $k \le n/2$, there are two matrices $A$, $B \in \mathcal{M}_n(\mathbb{C})$ with the property $A^2B = A$, such that $\mathrm{rank}(AB - BA) = k$.
$$
(AB - BA)^2 = O_n.
$$
b) Show that, for any natural number $k \le n/2$, there are two matrices $A$, $B \in \mathcal{M}_n(\mathbb{C})$ with the property $A^2B = A$, such that $\mathrm{rank}(AB - BA) = k$.