Answer:
Option D
Explanation:
Let $M=\begin{bmatrix}a_{1} & a_{2}&a_{3} \\b_{1} & b_{2}&b_{3}\\c_{1}&c_{2}&c_{3} \end{bmatrix}$
$\therefore$ $M\begin{bmatrix}0 \\1\\0 \end{bmatrix}=\begin{bmatrix}-1 \\2\\3 \end{bmatrix},M\begin{bmatrix}1 \\-1\\0 \end{bmatrix}=\begin{bmatrix}1 \\1\\-1 \end{bmatrix}$
$ M\begin{bmatrix}1 \\1\\1 \end{bmatrix}=\begin{bmatrix}0 \\0\\12 \end{bmatrix}$
$\Rightarrow$$ \begin{bmatrix}a_{2} \\b_{2}\\c_{2} \end{bmatrix}=\begin{bmatrix}-1 \\2\\3 \end{bmatrix},\begin{bmatrix}a_{1}-a_{2} \\b_{1}-b_{2}\\c_{1}-c_{2} \end{bmatrix}=\begin{bmatrix}1 \\1\\-1 \end{bmatrix}$
$,\begin{bmatrix}a_{1}+a_{2}+a_{3} \\b_{1}+b_{2}+b_{3}\\c_{1}+c_{2}+c_{3} \end{bmatrix}=\begin{bmatrix}0 \\0\\12 \end{bmatrix}$
$\Rightarrow$ $a_{2}=-1,b_{2}=2,c_{2}=3,a_{1}-a_{2}=1$, $b_{1}-b_{2}=1,c_{1}-c_{2}=-1$
$\Rightarrow$ $a_{1}+a_{2}+a_{3}=0,b_{1}+b_{2}+b_{3}=0,$
$c_{1}+c_{2}+c_{3}=12$
$\therefore$ $a_{1}=0$ ,$b_{2}=2$ and $c_{3}=7$
Hence, sum of diagonal elements
=0+2+7=9
