Let M be a $3 imes 3$ matrix satisfying $M\left[\begin{array}{l}0 \ 1 \ 0\end{array} ight]=\lef

Question

Let M be a $3 	imes 3$ matrix satisfying $M\left[\begin{array}{l}0 \ 1 \ 0\end{array}ight]=\left[\begin{array}{c}-1 \ 2 \ 3\end{array}ight], M\left[\begin{array}{c}1 \ -1 \ 0\end{array}ight]=\left[\begin{array}{c}1 \ 1 \ -1\end{array}ight]$ and $M\left[\begin{array}{l}1 \ 1 \ 1\end{array}ight]=\left[\begin{array}{c}0 \ 0 \ 12\end{array}ight]$. Then the sum of diagonal entries of $M$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

$M=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}\end{array}ight]$ Then $\quad a_{12}=-1$ $$ \begin{aligned} & a_{11}-a_{12}=1 \Rightarrow a_{11}=0 ; a_{11}+a_{12}+a_{13}=0 \Rightarrow a_{13}=1 \ & a_{22}=2, a_{21}-a_{22}=1 \Rightarrow a_{21}=3 ; a_{21}+a_{22}+a_{23}=0 \Rightarrow a_{21}=3 ; a_{21}+a_{22}+a_{23}=0 \Rightarrow a_{13}=-5 \ & a_{32}=3, a_{31}-a_{32}=1 \Rightarrow a_{31}=2 ; a_{31}+a_{32}+a_{33}=12 \Rightarrow a_{33}=7 \end{aligned} $$ Hence sum of diagonal of M is $a_{11}+a_{22}+a_{23}=0+2+7=9$