International Mathematics Competition
for University Students
2026

IMC2026: Day 1, Problem 2

Problem 2. Let \(\displaystyle n\) be a positive integer. Suppose that \(\displaystyle A\) and \(\displaystyle B\) are \(\displaystyle n\times n\) matrices with real entries such that

\(\displaystyle A^\top A + BB^\top = AB + BA,\)

where \(\displaystyle X^\top\) denotes the transpose of matrix \(\displaystyle X\).

Does this imply that \(\displaystyle AB = BA\)?

Nikolaos Kolliopoulos, University of Cyprus

  Solution