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

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

Answer is given below (on video)