International Mathematics Competition
for University Students
2019

IMC2019: Day 2, Problem 9

Problem 9. Determine all positive integers \(\displaystyle n\) for which there exist \(\displaystyle n\times n\) real invertible matrices \(\displaystyle A\) and \(\displaystyle B\) that satisfy \(\displaystyle AB-BA=B^2A\).

Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

Hint: If \(\displaystyle n\) is odd, consider the real eigenvalues of \(\displaystyle B\) and \(\displaystyle B^2+B\).

If \(\displaystyle n\) is even, build a block matrix from \(\displaystyle 2\times2\) blocks.

  Solution