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A^2002=B^2003=I and AB=BA implies A+B+I invertible

Source: VJIMC 2003 2.1

July 13, 2021

matrixlinear algebra

Problem Statement

Two real square matrices

A

and

B

satisfy the conditions

A^{2002}=B^{2003}=I

and

AB=BA

. Prove that

A+B+I

is invertible. (The symbol

I

denotes the identity matrix.)