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

\(A=\begin{bmatrix}a & \frac{q(a-d)}{p-s} \\ \frac{r(a-d)}{p-s} & d \end{bmatrix}\)

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

\(A=\begin{bmatrix}a & \frac{q(a-d)}{p-s} \\ \frac{r(a-d)}{p-s} & d \end{bmatrix}\)