International Mathematics Competition
for University Students
2023

IMC2023: Day 1, Problem 2

Problem 2. Let \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) be \(\displaystyle n\times n\) matrices with complex entries satisfying

\(\displaystyle A^2 = B^2 = C^2 \quad \text{and} \quad B^3 = ABC + 2I. \)

Prove that \(\displaystyle A^6 = I\).

Mike Daas, Universiteit Leiden

Hint: Factorize \(\displaystyle B^3-ABC\).

  Solution