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Romanian National Olympiad 2024 - Grade 11 - Problem 2

Source: Romanian National Olympiad 2024 - Grade 11 - Problem 2

April 4, 2024
linear algebramatrix

Problem Statement

Let A∈Mn(R)A \in \mathcal{M}_n(\mathbb{R}) be an invertible matrix.
a) Prove that the eigenvalues of AATAA^T are positive real numbers. b) We assume that there are two distinct positive integers, pp and qq, such that (AAT)p=(ATA)q.(AA^T)^p=(A^TA)^q. Prove that AT=A−1.A^T=A^{-1}.
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