IMC 2019 · Problem 5

Determine whether there exist an odd positive integer $n$ and $n \times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:

(1) $\det(B) = 1$;

(2) $AB = BA$;

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

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