IMC 2007 · Problem 5

For each positive integer $k$, find the smallest number $n_k$ for which there exist real $n_k \times n_k$ matrices $A_1, A_2, \dots, A_k$ such that all of the following conditions hold:

(1) $A_1^2 = A_2^2 = \dots = A_k^2 = 0$,

(2) $A_i A_j = A_j A_i$ for all $1 \le i, j \le k$, and

(3) $A_1 A_2 \dots A_k \ne 0$.