Question

Let $A=\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$. Then the number of $3 \times 3$ matrices $B$ with entries from the set $\{1,2,3,4,5\}$ and satisfying $A B=B A$ is_______.

Answer 1

Let matrix $B=\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$
$\because A B=B A$
$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}=\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} $
$\begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \end{bmatrix}=\begin{bmatrix} b & a & c \\ e & d & f \\ h & g & i \end{bmatrix}$
$\Rightarrow d=b, e=a, f=c, g=h$
$\therefore $ Matrix $B=\begin{bmatrix}a & b & c \\ b & a & c \\ g & g & i\end{bmatrix}$
No. of ways of selecting $a, b, c, g, i$
$=5 \times 5 \times 5 \times 5 \times 5$
$=5^{5}=3125$
$\therefore $ No. of Matrices $B=3125$