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Matrices satisfy three conditions

Source: IMC 2019 Day 1 P5

July 31, 2019
linear algebracollege contests

Problem Statement

Determine whether there exist an odd positive integer nn and n×nn\times n matrices AA and BB with integer entries, that satisfy the following conditions:
[*]det⁡(B)=1\det (B)=1;[/*] [*]AB=BAAB=BA;[/*] [*]A4+4A2B2+16B4=2019IA^4+4A^2B^2+16B^4=2019I.[/*]
(Here II denotes the n×nn\times n identity matrix.)
Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan
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