Question

Let $M$ be a $3 \times 3$ 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 1

Let $M=\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$
$M \begin{bmatrix}0 \\1 \\0\end{bmatrix}=\begin{bmatrix}-1 \\2 \\3\end{bmatrix}$
$\Rightarrow b =-1, e =2,\,\,h =3$
$M \begin{bmatrix}1 \\-1 \\0\end{bmatrix}=\begin{bmatrix}1 \\ 1 \\-1\end{bmatrix}$
$\Rightarrow a =0, \,\,d =3, \,\,g =2$
$M \begin{bmatrix}1 \\1 \\1 \end{bmatrix}=\begin{bmatrix}0 \\0 \\12\end{bmatrix}$
$\Rightarrow g + h + i =12 \Rightarrow i =7$
$\therefore $ Sum of diagonal elements $=9$