International Mathematics Competition
for University Students
2024

IMC2024: Day 1, Problem 3

Problem 3. For which positive integers \(\displaystyle n\) does there exist an \(\displaystyle n\times n\) matrix \(\displaystyle A\) whose entries are all in \(\displaystyle \{0,1\}\), such that \(\displaystyle A^2\) is the matrix of all ones?

Alex Avdiushenko, Neapolis University Paphos, Cyprus

Hint: Let \(\displaystyle J\) be the \(\displaystyle n\times n\) matrix with all ones. Consoider \(\displaystyle A^3=AJ=JA\).

  Solution