Algebra Solutions: Demonstrate that a given set of matrices is closed under matrix addition

Denote by $\mathcal{A}$ the set of all 2×2 matrices with real number entries. Let $M = \left[ {1 \atop 0} {1 \atop 1} \right]$ and let $\mathcal{B} = \{ X \in \mathcal{A} \ |\ MX = XM \}$ .

Prove that if $P, Q \in \mathcal{B}$ , then $P+Q \in \mathcal{B}$ , where + denotes the usual sum of two matrices.

Recall that matrix multiplication distributes over matrix addition on both sides. Thus $(P+Q)M = PM + QM = MP + MQ$ , since $P,Q \in \mathcal{B}$ , and so $(P+Q)M = M(P+Q)$ . $\blacksquare$

Algebra Solutions

Demonstrate that a given set of matrices is closed under matrix addition

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