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Omni-MATH / Let n be a positive integer. What is the largest k for which there exist n×n matrices M₁,…,M_(k) and N₁,…,N_(k) with real entries such that for all i and j, the matrix product …

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problem

Let nn be a positive integer. What is the largest kk for which there exist n×nn \times n matrices M1,…,MkM_1, \dots, M_k and N1,…,NkN_1, \dots, N_k with real entries such that for all ii and jj, the matrix product MiNjM_i N_j has a zero entry somewhere on its diagonal if and only if i≠ji \neq j?
Plain-text mathematical notation (without MathML)
Let n be a positive integer. What is the largest k for which there exist n×n matrices M₁,…,M_(k) and N₁,…,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≠j?
Original LaTeX notation
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 $i \neq j$?

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task_id
887f4f56-f3c8-5585-883a-36fe0b7feab9
task_key
test--887f4f56-f3c8-5585-883a-36fe0b7feab9
task_revision_id
1
config / split
/ test

Official source

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