Mathematics - Matrices Question with Solution | TestHub

Let $M=\left[\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$ $\therefore M\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{r}-1\\ 2\\ 3\end{array}\right],M\left[\begin{array}{r}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{r}1\\ 1\\ -1\end{array}\right]$, $M\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{r}0\\ 0\\ 12\end{array}\right]$, $$\begin{array}{rl}\Rightarrow & \left[\begin{array}{l}{a}_2\\ b_2\\ c_2\end{array}\right]=\left[\begin{array}{r}-1\\ 2\\ 3\end{array}\right],\left[\begin{array}{l}{a}_1-a_2\\ b_1-b_2\\ c_1-c_2\end{array}\right]=\left[\begin{array}{r}1\\ 1\\ -1\end{array}\right],\\ & \left[\begin{array}{l}{a}_1+a_2+a_3\\ b_1+b_2+b_3\\ c_1+c_2+c_3\end{array}\right]=\left[\begin{array}{r}0\\ 0\\ 12\end{array}\right]\\ \Rightarrow & a_2=-1,b_2=2,c_2=3,a_1-a_2=1,\\ \Rightarrow & b_1-b_2=1,c_1-c_2=-1\\ \Rightarrow & a_1+a_2+a_3=0,b_1+b_2+b_3=0,\\ \therefore & c_1+c_2+c_3=12\\ \therefore & a_1=0,b_2=2\text{ and }{c}_3=7\end{array}$$ Hence, sum of diagonal elements $$=0+2+7=9$$