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given p^2*A^(p^2)=q^2*A^(q^2)+r^2*I_n

Source: VJIMC 2009 2.3

June 12, 2021
matrixlinear algebra

Problem Statement

Let AA be an n×nn\times n square matrix with integer entries. Suppose that p2Ap2=q2Aq2+r2Inp^2A^{p^2}=q^2A^{q^2}+r^2I_n for some positive integers p,q,rp,q,r where rr is odd and p2=q2+r2p^2=q^2+r^2. Prove that ∣det⁡A∣=1|\det A|=1. (Here InI_n means the n×nn\times n identity matrix.)
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