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Finding det(AB) if A+B=I and (A^2+B^2)(A^4+B^4)=A^5+B^5

Source: IMC 2024, Problem 7

August 8, 2024

linear algebradeterminantMatrices

Problem Statement

Let

n

be a positive integer. Suppose that

A

and

B

are invertible

n \times n

matrices with complex entries such that

A+B=I

(where

I

is the identity matrix) and

(A^2+B^2)(A^4+B^4)=A^5+B^5.

Find all possible values of

\det(AB)

for the given

n