International Mathematics Competition
for University Students
2024

IMC2024: Day 2, Problem 7

Problem 7. Let \(\displaystyle n\) be a positive integer. Suppose that \(\displaystyle A\) and \(\displaystyle B\) are invertible \(\displaystyle n\times n\) matrices with complex entries such that \(\displaystyle A+B=I\) (where \(\displaystyle I\) is the identity matrix) and

\(\displaystyle (A^2 + B^2)(A^4 + B^4) = A^5 + B^5. \)

Find all possible values of \(\displaystyle \det(AB)\) for the given \(\displaystyle n\).

Sergey Bondarev, Sergey Chernov, Belarusian State University, Minsk

Hint: Find a polynomial \(\displaystyle p(x)\) such that \(\displaystyle p(AB)=0\).

  Solution