Question

Let $A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right].$ Then the number of $3\times 3$ matrices $B$ with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is

Answer 1

Let $B=\left[\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right]$

$AB=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right]=\left[\begin{array}{ccc}{\beta }_1& {\beta }_2& {\beta }_3\\ {\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right]$

$BA=\left[\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_2& {\alpha }_1& {\alpha }_3\\ {\beta }_2& {\beta }_1& {\beta }_3\\ {\gamma }_2& {\gamma }_1& {\gamma }_3\end{array}\right]$

$AB=BA\Rightarrow {\beta }_1={\alpha }_2,{\beta }_2={\alpha }_1,{\beta }_3={\alpha }_3,{\gamma }_1={\gamma }_2$
$5$ places can be filled independently in $5^5=3125$ ways $=3125$ matrices