International Mathematics Competition
for University Students
2019

IMC2019: Day 1, Problem 5

Problem 5. Determine whether there exist an odd positive integer \(\displaystyle n\) and \(\displaystyle n\times n\) matrices \(\displaystyle A\) and \(\displaystyle B\) with integer entries, that satisfy the following conditions:

    (1) \(\displaystyle \det(B)=1\);

    (2) \(\displaystyle AB=BA\);

    (3) \(\displaystyle A^4+4A^2B^2+16B^4=2019I\).

(Here \(\displaystyle I\) denotes the \(\displaystyle n\times n\) identity matrix.)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

Hint: Consider the determinants modulo \(\displaystyle 4\).

  Solution