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: [list=1] [] $\det (B)=1$ ;[/] [] $AB=BA$ ;[/] [] $A^4+4A^2B^2+16B^4=2019I$ .[/] [/list] (Here $I$ denotes the $n\times n$ identity matrix.)

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