Discussion Forum

1)

Let M be a 3 x3 matrix satisfying

$M\begin{bmatrix}0  \\1 \\0 \end{bmatrix}=\begin{bmatrix}-1  \\2 \\3 \end{bmatrix},M\begin{bmatrix}1  \\-1\\0 \end{bmatrix}=\begin{bmatrix}1  \\1 \\-1 \end{bmatrix}$  and  $M\begin{bmatrix}1  \\1 \\1 \end{bmatrix}=\begin{bmatrix}0  \\0 \\12 \end{bmatrix}$

Then, the sum  of the diagonal entries of M is

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