Question

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

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

2021-03-05
Step 1
Let $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix} \text{ and } B=\begin{bmatrix}p & q \\ r & s \end{bmatrix}$$ be any $$2 \times 2$$ matrices.
Step 5
Find AB.
$$AB=\begin{bmatrix}a & b \\c & d \end{bmatrix}\begin{bmatrix}p & q \\ r & s \end{bmatrix}$$
$$=\begin{bmatrix}ap+br & aq+bs \\ cp+dr & cq+ds \end{bmatrix}$$
Find BA.
$$BA=\begin{bmatrix}p & q \\ r & s \end{bmatrix}\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
$$=\begin{bmatrix}ap+cq & bp+dq \\ ar+cs & br+ds \end{bmatrix}$$ Equate the matrices AB=BA.
AB=BA
$$\begin{bmatrix}ap+br & aq+bs \\ cp+dr & cq+ds \end{bmatrix}=\begin{bmatrix}ap+cq & bp+dq \\ ar+cs & br+ds \end{bmatrix}$$
$$\begin{bmatrix}ap+br & aq+bs \\ cp+dr & cq+ds \end{bmatrix}-\begin{bmatrix}ap+cq & bp+dq \\ ar+cs & br+ds \end{bmatrix}= \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}$$
$$\begin{bmatrix}br-cq & (a-d)q-b(p-s) \\ c(p-s)-r(a-d) & cq-br \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}$$
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
$$A=\begin{bmatrix}a & \frac{q(a-d)}{p-s} \\ \frac{r(a-d)}{p-s} & d \end{bmatrix}$$