Let $A=\left[a_{i j} ight]$ be $3 imes 3$ matrix such that $A\left[\begin{array}{l}0 \ 1 \ 0\end{array} ight]=\left[\begin{array}{l}0 \ 0 \ 1\end{array} ight], A\left[\begin{array}{l}4 \ 1 \ 3\end{array} ight]=\left[\begin{array}{l}0 \ 1 \ 0\end{array} ight]$ and…

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Let $A=\left[a_{i j}ight]$ be $3 	imes 3$ matrix such that $A\left[\begin{array}{l}0 \ 1 \ 0\end{array}ight]=\left[\begin{array}{l}0 \ 0 \ 1\end{array}ight], A\left[\begin{array}{l}4 \ 1 \ 3\end{array}ight]=\left[\begin{array}{l}0 \ 1 \ 0\end{array}ight]$ and $A\left[\begin{array}{l}2 \ 1 \ 2\end{array}ight]=\left[\begin{array}{l}1 \ 0 \ 0\end{array}ight]$, then $a_{23}$ equals :

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Let } A=\left[\begin{array}{lll} a & b & c \ d & e & f \ g & h & i \end{array}ight] \ & 	herefore\left[\begin{array}{lll} a & b & c \ d & e & f \ g & h & i \end{array}ight]\left[\begin{array}{l} 0 \ 1 \ 0 \end{array}ight]=\left[\begin{array}{l} 0 \ 0 \ 1 \end{array}ight] \ & 	herefore \quad b=0, e=0, h=1 \ & 	ext { and }\left[\begin{array}{lll} a & 0 & c \ d & 0 & f \ g & 1 & i \end{array}ight]\left[\begin{array}{l} 4 \ 1 \ 3 \end{array}ight]=\left[\begin{array}{l} 0 \ 1 \ 0 \end{array}ight] \ & \left.	herefore \begin{array}{l} 4 a+3 c=0 \ 4 d+3 f=1 \ 4 g+1+3 i=0 \end{array}ight\} ....(1) \end{aligned}$ $\begin{aligned} & 	ext { and }\left[\begin{array}{ccc}a & 0 & c \ d & 0 & f \ g & 1 & i\end{array}ight]\left[\begin{array}{l}2 \ 1 \ 2\end{array}ight]=\left[\begin{array}{l}1 \ 0 \ 0\end{array}ight] \ & \left.\begin{array}{c}2 a+2 c=1 \ 	herefore 2 d+2 f=0 \ 2 g+1+2 i=0\end{array}ight\}...(2)\end{aligned}$ From equation (1) and (2) we get $\begin{aligned} & d=1, f=-1 \ & 	herefore \quad a_{23}=-1 \end{aligned}$ ,

Let $A=\left[a_{i j}\right]$ be $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\…

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