QJEEM25Mafffa: JEE Main 2025 Mathematics - Class 12 JEEM25Mafffa Solution: Let be a matrix such that and , then equals :

$\begin{aligned}& \text { Let } A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right] \\& A\left[\begin{array}{l}0 \\1 \\0\end{array}\right]=\left[\begin{array}{l}0 \\0 \\1\end{array}\right] \Rightarrow\left[\begin{array}{l}a_{12} \\a_{22} \\a_{32}\end{array}\right]=\left[\begin{array}{l}0 \\1 \\0\end{array}\right] \Rightarrow \begin{array}{l}a_{22}=0 ; a_{12}=0 \\a_{32}=1\end{array}\end{aligned}$

$\begin{aligned}A\left[\begin{array}{l}4 \\1 \\3\end{array}\right]=\left[\begin{array}{l}0 \\1 \\0\end{array}\right] \Rightarrow \begin{array}{l}4 a_{11}+a_{12}+3 a_{13}=0 \\4 a_{21}+a_{22}+3 a_{23}=1 \Rightarrow 4 a_{21}+3 a_{23}=1 \\4 a_{31}+a_{32}+3 a_{33}=0\end{array} \\A\left[\begin{array}{l}2 \\1 \\2\end{array}\right]=\left[\begin{array}{l}1 \\0 \\0\end{array}\right] \Rightarrow \begin{array}{l}2 a_{11}+a_{12}+2 a_{13}=1 \\2 a_{21}+a_{22}+2 a_{23}=0 \Rightarrow a_{21}+a_{23}=0 \\2 a_{31}+a_{32}+2 a_{33}=0\end{array} \\-4 a_{23}+3 a_{23}=1 \Rightarrow a_{23}=-1\end{aligned}$