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