Josalynn

2021-03-04

Determine all $2×2$ matrices A such that AB = BA for any $2×2$ matrix B.

hesgidiauE

Step 1
Let be any $2×2$ matrices.
Step 5
Find AB.
$AB=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$
$=\left[\begin{array}{cc}ap+br& aq+bs\\ cp+dr& cq+ds\end{array}\right]$
Find BA.
$BA=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$=\left[\begin{array}{cc}ap+cq& bp+dq\\ ar+cs& br+ds\end{array}\right]$ Equate the matrices AB=BA.
AB=BA
$\left[\begin{array}{cc}ap+br& aq+bs\\ cp+dr& cq+ds\end{array}\right]=\left[\begin{array}{cc}ap+cq& bp+dq\\ ar+cs& br+ds\end{array}\right]$
$\left[\begin{array}{cc}ap+br& aq+bs\\ cp+dr& cq+ds\end{array}\right]-\left[\begin{array}{cc}ap+cq& bp+dq\\ ar+cs& br+ds\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\left[\begin{array}{cc}br-cq& \left(a-d\right)q-b\left(p-s\right)\\ c\left(p-s\right)-r\left(a-d\right)& cq-br\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Equate the matrices.
$br-cq=0\dots \left(1\right)$
$\left(a-d\right)q-b\left(p-s\right)=0\dots \left(2\right)$
$c\left(p-s\right)-r\left(a-d\right)=0\dots \left(3\right)$
$cq-br=0\dots \left(4\right)$
From equation (1) and equation (4),
$q=\frac{b}{c}r$
Substitute $q=\frac{b}{c}r$ in equation (2)
$\left(a-d\right)\frac{b}{c}e-b\left(p-s\right)=0$
$c=\frac{\left(a-d\right)r}{\left(p-s\right)}$
Substitute $q=\frac{b}{c}r$ in equation (2)
$\left(a-d\right)\left(\frac{b}{c}r\right)-b\left(p-s\right)=0$
$b=\frac{q\left(a-d\right)}{p-s}$
Hence, the 2x2 matrix which satisfied AB=BA for any matrix B is
$A=\left[\begin{array}{cc}a& \frac{q\left(a-d\right)}{p-s}\\ \frac{r\left(a-d\right)}{p-s}& d\end{array}\right]$

Jeffrey Jordon