Answer
Matrix A is $A=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]$ and matrix B is $B=\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]$.
Work Step by Step
Let,
$A=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]$
And,
$B=\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]$
Then, the product operation on matrices is performed as shown below:
$\begin{align}
& AB=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1\left( a \right)+0\left( b \right) & 1\left( c \right)+0\left( d \right) \\
0\left( a \right)+1\left( b \right) & 0\left( c \right)+1\left( d \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]
\end{align}$
Then,
$\begin{align}
& BA=\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
a\left( 1 \right)+c\left( 0 \right) & a\left( 0 \right)+c\left( 1 \right) \\
b\left( 1 \right)+d\left( 0 \right) & b\left( 0 \right)+d\left( 1 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]
\end{align}$
Therefore, the two matrices satisfy the condition $AB=BA$.
Therefore, the required matrices are $A=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]$ and $B=\left[ \begin{matrix}
a & c \\
b & d \\
\end{matrix} \right]$.
