JEE Main 2025 — Matrices Question with Solution

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