MathDB

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}, \ldots, A_{k}

such that all of the following conditions hold: (1)

A_{1}^{2}= A_{2}^{2}= \ldots = 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}\ldots A_{k}\ne 0

Problem Statement