MathNet 2017-part-05 — 176 problems (EN–VI) Question 62

Let $A$, $B \in \mathcal{M}_n(\mathbb{C})$ be matrices such that $A + B$ is invertible and $A^2 = B^2 = O_n$. Show that $n$ is even and the matrices $(AB)^k$, $k = 1, 2, \dots$, have rank $\frac{n}{2}$.

Florin Stănescu