Let M be a 3x3 matrix satisfying
\[M\left( {\begin{array}{*{20}{c}}
  0 \\
  1 \\
  0
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  { - 1} \\
  2 \\
  3
\end{array}} \right)\] , $M\left( {\begin{array}{*{20}{c}}
  1 \\
  { - 1} \\
  0
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  { - 1}
\end{array}} \right)$ and $M\left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  1
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  {12}
\end{array}} \right)$ .
Then the sum of diagonal entries of M is

Hint: There are three given equations, and M is a 3x3 matrix. First consider the elements of as some variable and expand equations with these elements of M and we get some simple algebraic equations in the terms of elements of matrix M. Using these algebraic equations calculate all elements of M. To find sum of diagonal entries add up all the elements present on diagonal of matrix M.

Complete step-by-step answer:
Let $M = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\
  {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\
  {{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\end{array}} \right)$
Given \[M\left( {\begin{array}{*{20}{c}}
  0 \\
  1 \\
  0
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  { - 1} \\
  2 \\
  3
\end{array}} \right)\] -(i),
$M\left( {\begin{array}{*{20}{c}}
  1 \\
  { - 1} \\
  0
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  { - 1}
\end{array}} \right)$ -(ii) and
 $M\left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  1
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  {12}
\end{array}} \right)$ -(iii).
By expanding eqn. (i), we get
\[\left( {\begin{array}{*{20}{c}}
  {{a_{11}}.0 + {a_{12}}.1 + {a_{13}}.0} \\
  {{a_{21}}.0 + {a_{22}}.1 + {a_{23}}.0} \\
  {{a_{31}}.0 + {a_{32}}.1 + {a_{33}}.0}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  { - 1} \\
  2 \\
  3
\end{array}} \right)\] or $\left( {\begin{array}{*{20}{c}}
  {{a_{12}}} \\
  {{a_{22}}} \\
  {{a_{32}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  { - 1} \\
  2 \\
  3
\end{array}} \right)$
Then ${a_{12}} = - 1$ , ${a_{22}} = 2$ and ${a_{32}} = 3$.
Then expand eqn. (ii) and we get
\[\left( {\begin{array}{*{20}{c}}
  {{a_{11}}.1 + {a_{12}}. - 1 + {a_{13}}.0} \\
  {{a_{21}}.1 + {a_{22}}. - 1 + {a_{23}}.0} \\
  {{a_{31}}.1 + {a_{32}}. - 1 + {a_{33}}.0}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  { - 1}
\end{array}} \right)\] or \[\left( {\begin{array}{*{20}{c}}
  {{a_{11}} - {a_{12}}} \\
  {{a_{21}} - {a_{22}}} \\
  {{a_{31}} - {a_{32}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  1 \\
  1 \\
  { - 1}
\end{array}} \right)\]
${a_{11}} - {a_{12}} = 1$ , ${a_{11}} = 1 + {a_{12}}$ , then ${a_{11}} = 1 - 1 = 0$.
${a_{21}} - {a_{22}} = 1$ , ${a_{21}} = 1 + {a_{22}}$ , then ${a_{21}} = 1 + 2 = 3$.
${a_{31}} - {a_{32}} = - 1$ , ${a_{31}} = - 1 + {a_{s2}}$ , then ${a_{31}} = - 1 + 3 = 2$.
Similarly, we expand eqn. (iii) and we get
\[\left( {\begin{array}{*{20}{c}}
  {{a_{11}}.1 + {a_{12}}.1 + {a_{13}}.1} \\
  {{a_{21}}.1 + {a_{22}}.1 + {a_{23}}.1} \\
  {{a_{31}}.1 + {a_{32}}.1 + {a_{33}}.1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  {12}
\end{array}} \right)\] or $\left( {\begin{array}{*{20}{c}}
  {{a_{11}} + {a_{12}} + {a_{13}}} \\
  {{a_{21}} + {a_{22}} + {a_{23}}} \\
  {{a_{31}} + {a_{32}} + {a_{33}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  {12}
\end{array}} \right)$
${a_{13}} = - {a_{11}} - {a_{12}}$, then ${a_{13}} = 0 + 1 = 1$.
${a_{23}} = - {a_{21}} - {a_{22}}$, then ${a_{23}} = - 3 - 2 = - 5$.
${a_{33}} = 12 - {a_{31}} - {a_{32}}$, then ${a_{33}} = 12 - 2 - 3 = 7$.
Here we get $M = \left( {\begin{array}{*{20}{c}}
  0&{ - 1}&1 \\
  3&2&{ - 5} \\
  2&3&7
\end{array}} \right)$. We have to find sum of the diagonal entries which is gives by
${a_{11}} + {a_{22}} + {a_{33}} = 0 + 2 + 7 = 9$

Then the correct answer is option A.

Note: When we equate two matrices like A=B then dimension of both matrices are equal and element of matrix A ${a_{ij}}$ is equal to some element of B ${b_{ij}}$, Such as ${a_{11}} = {b_{11}}$. This property helps us to find the matrix M when M is equated to some given matrix by equating element on left to element on right side as the same position.
Here we find the complete matrix M, you can also find only the element needed to find the answer.