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M2(R), existence of matrices satisfying conditions

Source: SEEMOUS 2010 P3

June 17, 2021
matrixlinear algebra

Problem Statement

Denote by M2(R)\mathcal M_2(\mathbb R) the set of all 2×22\times2 matrices with real entries. Prove that:
a) for every A∈M2(R)A\in\mathcal M_2(\mathbb R) there exist B,C∈M2(R)B,C\in\mathcal M_2(\mathbb R) such that A=B2+C2A=B^2+C^2; b) there do not exist B,C∈M2(R)B,C\in\mathcal M_2(\mathbb R) such that (0110)=B2+C2\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2 and BC=CBBC=CB.
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