Let $A=\left[\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1\end{array} ight]$. Then the numb

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

Professor PC Thomas Classes · Answer

(3125) : Let $B=\left[\begin{array}{ccc}p & q & r \ s & t & u \ v & w & x\end{array}ight]$ be $3 	imes 3$ matrix such that $A B=B A$\ $\Rightarrow\left[\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1\end{array}ight]\left[\begin{array}{lll}p & q & r \ s & t & u \ v & w & x\end{array}ight]$\ $=\left[\begin{array}{lll}p & q & r \ s & t & u \ v & w & x\end{array}ight]\left[\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1\end{array}ight]$\ $\Rightarrow\left[\begin{array}{lll}s & t & u \ p & q & r \ v & w & x\end{array}ight]=\left[\begin{array}{lll}q & p & r \ t & s & u \ w & v & x\end{array}ight]$\ $\Rightarrow s=q, t=p, u=r, v=w$\ So, we have to select 5 elements for $B$.\ $\because \quad$ Each element of $B$ have five options 1 , $2,3,4,5$.\ $	herefore \quad$ Total number of ways $=5^{5}=3125$.\