Algebra Solutions: Characterize a given set of matrices in terms of their entries

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

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, (1) $p = p+r$ , (2) $p+q = q+s$ , (3) $r = r$ , and (4) $r+s = s$ .
From (1) or (4) we deduce that $r = 0$ , and from (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.

Algebra Solutions

Characterize a given set of matrices in terms of their entries

No comments:

Post a Comment