IMC 2022 · Problem 7

Let $A_1, A_2, \ldots, A_k$ be $n \times n$ idempotent complex matrices such that

$$A_iA_j = -A_jA_i \quad \text{for all } i \ne j.$$

Prove that at least one of the given matrices has rank $\le \frac{n}{k}$.

(A matrix $A$ is called idempotent if $A^2 = A$.)