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Demonstrate that a given set of matrices is multiplicatively closed

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.3

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 $PQ \in \mathcal{B}$, where juxtaposition denotes the usual matrix product.

Solution: Recall that matrix multiplication is associative. Since $P,Q \in \mathcal{B}$, we have $PM=MP$ and $QM=MQ$. Then we have $$(PQ)M = P(QM) = P(MQ) = (PM)Q = (MP)Q = M(PQ),$$ and thus $PQ \in \mathcal{B}$.


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