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Integer matrices known fact

Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 4

October 21, 2023
matrixMatrix algebralinear algebraeigenvalue and eigenvector

Problem Statement

Let M2(Z)M_2(\mathbb{Z}) be the set of 2×22 \times 2 matrices with integer entries. Let A∈M2(Z)A \in M_2(\mathbb{Z}) such that A2+5I=0,A^2+5I=0, where I∈M2(Z)I \in M_2(\mathbb{Z}) and 0∈M2(Z)0 \in M_2(\mathbb{Z}) denote the identity and null matrices, respectively. Prove that there exists an invertible matrix C∈M2(Z)C \in M_2(\mathbb{Z}) with C−1∈M2(Z)C^{-1} \in M_2(\mathbb{Z}) such that CAC−1=(12−3−1) ou CAC−1=(01−50).CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.
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