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Characterize a given set of matrices in terms of their entries


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.4

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 $$\mathcal{B} = \{ X \in \mathcal{A} \ |\ MX = XM \}.$$ Find conditions on $p, q, r, s$ which determine precisely when $$A = \left[ {p \atop r} {q \atop s} \right] \in \mathcal{B}.$$


Solution: Recall that two matrices are equal precisely when their corresponding entries are equal. We have $$A M = \left[ {p \atop r}{{p+q} \atop {r+s}} \right]$$ and $$M A = \left[ {{p+r} \atop r}{{q+s} \atop s} \right];$$ if we demand that these be equal, we get a system of four equations in four unknowns. Namely, \begin{align}&p = p+r,\label{eq:1}\\ &p+q = q+s,\label{eq:2}\\ & r = r,\\& r+s = s\label{eq:4}.\end{align}

From \eqref{eq:1} or \eqref{eq:4} we deduce that $r = 0$, and from \eqref{eq:2} that $p = s$. Thus an arbitrary element of $\mathcal{B}$ is of the form $$\left[ {p \atop 0}{q \atop p} \right]$$ for some $p,q$. Moreover, every matrix of this form is in $\mathcal{B}$, as is easily verified.


Linearity

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