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

asked 2021-03-04
Determine all \(2 \times 2\) matrices A such that AB = BA for any \(2 \times 2\) matrix B.

Answers (1)

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.
\(\begin{bmatrix}ap+br & aq+bs \\ cp+dr & cq+ds \end{bmatrix}=\begin{bmatrix}ap+cq & bp+dq \\ ar+cs & br+ds \end{bmatrix}\)
\(\begin{bmatrix}ap+br & aq+bs \\ cp+dr & cq+ds \end{bmatrix}-\begin{bmatrix}ap+cq & bp+dq \\ ar+cs & br+ds \end{bmatrix}= \begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}\)
\(\begin{bmatrix}br-cq & (a-d)q-b(p-s) \\ c(p-s)-r(a-d) & cq-br \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}\)
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),
Substitute \(q=\frac{b}{c}r\) in equation (2)
\(c= \frac{(a-d)r}{(p-s)}\)
Substitute \(q=\frac{b}{c}r\) in equation (2)
Hence, the 2x2 matrix which satisfied AB=BA for any matrix B is
\(A=\begin{bmatrix}a & \frac{q(a-d)}{p-s} \\ \frac{r(a-d)}{p-s} & d \end{bmatrix}\)
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