IMC 2016 · Problem 2

Let $k$ and $n$ be positive integers. A sequence $(A_1, \ldots, A_k)$ of $n \times n$ real matrices is preferred by Ivan the Confessor if $A_i^2 \ne 0$ for $1 \le i \le k$, but $A_iA_j = 0$ for $1 \le i, j \le k$ with $i \ne j$. Show that $k \le n$ in all preferred sequences, and give an example of a preferred sequence with $k = n$ for each $n$.