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matrix A^{3n} takes only two values if A^2+I=A

Source: 1988 Greece MO Grade XII p3

September 7, 2024
linear algebramatrix

Problem Statement

Let AA be a n×nn \times n matrix of real numbers such that A2+I=AA^2+\mathbb{I}=A, where I\mathbb{I} is the identity n×nn \times n matrix. Prove that the matrix A3nA^{3n} , where ν∈Z\nu\in\mathbb{Z} takes only two values and find those values.
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