Mathematics - Matrices Question with Solution | TestHub

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