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


Denote by $\mathcal{A}$ the set of all $2\times 2$ 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.


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


Linearity

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