International Mathematics Competition
for University Students
2021

IMC2021: Day 2, Problem 5

Problem 5. Let \(\displaystyle A\) be a real \(\displaystyle n\times n\) matrix and suppose that for every positive integer \(\displaystyle m\) there exists a real symmetric matrix \(\displaystyle B\) such that

\(\displaystyle 2021B=A^m+B^2.\)

Prove that \(\displaystyle |\det{A}|\le 1\).

Rafael Filipe dos Santos, Instituto Militar de Engenharia, Rio de Janeiro

Hint: The determinant is the product of the eigenvalues.

  Solution