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IMC 2021 P5: 2021B=A^m+B^2

Source: IMC 2021 P5

August 5, 2021
linear algebraIMC 2021matrixsymmetric matrix

Problem Statement

Let AA be a real n×nn \times n matrix and suppose that for every positive integer mm there exists a real symmetric matrix BB such that
2021B=Am+B2.2021B = A^m+B^2.
Prove that ∣detA∣≤1|\text{det} A| \leq 1.
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