Putnam 2014 A6

Let $n$ be a positive integer. What is the largest $k$ for which there exist $n \times n$ matrices $M_1, \dots, M_k$ and $N_1, \dots, N_k$ with real entries such that for all $i$ and $j$, the matrix product $M_i N_j$ has a zero entry somewhere on its diagonal if and only if $i \neq j$?