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

Question

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. $A^4+4A^2B^2+16B^4=2019I$ (Where $I$ is the $n \times n$ identity matrix.)

I know that $(A^2+4B^2+2AB)(A^2+4B^2-2AB)=2019I$. From this how to proceed?

Answer 1

See here : (it's the problem 5 from IMC 2019)

https://www.imc-math.org.uk/?year=2019&section=problems&item=prob5s