Question

Let M be a $3 \times 3$ 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 the diagonal entries of M is ____.

Answer 1

Hint : To solve this question, we will start with assuming the $3 \times 3$ matrix. Now we have been given three conditions which satisfies the matrix M. So, one by one we will take the condition, and put the value of assumed matrix M, then on first condition, we will get the value of three entries of the matrix, on solving the second condition, we will get the values of another 3 entries of matrix, similarly on solving the third condition, we will get the last three entries of matrix, hence after getting all the entries of the matrix, we will take the sum of one of the diagonal of matrix, and hence we will get our required answer.

Complete step by step solution:
We have been given that M is a $3 \times 3$ matrix. So, let M = \left[ {\begin{array}{*{20}{c}} a&b;&c; \\\ d&e;&f; \\\ g&h;&i; \end{array}} \right]
Now, according to the question, M satisfies,

0 \\\ 1 \\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1} \\\ 2 \\\ 3 \end{array}} \right]$$

\left[ {\begin{array}{{20}{c}}
a&b;&c; \\
d&e;&f; \\
g&h;&i;
\end{array}} \right]\left[ {\begin{array}{{20}{c}}
0 \\
1 \\
0
\end{array}} \right] = \left[ {\begin{array}{{20}{c}}
{ - 1} \\
2 \\
3
\end{array}} \right] \\
\left[ {\begin{array}{{20}{c}}
b \\
e \\
h
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{ - 1} \\
2 \\
3
\end{array}} \right] \\
b = - 1,e = 2,h = 3. \\

It is also given that, M satisfies, $$M\left[ {\begin{array}{*{20}{c}} 1 \\\ { - 1} \\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1 \\\ 1 \\\ { - 1} \end{array}} \right]$$

\left[ {\begin{array}{{20}{c}}
a&b;&c; \\
d&e;&f; \\
g&h;&i;
\end{array}} \right]\left[ {\begin{array}{{20}{c}}
1 \\
{ - 1} \\
0
\end{array}} \right] = \left[ {\begin{array}{{20}{c}}
1 \\
1 \\
{ - 1}
\end{array}} \right] \\
\left[ {\begin{array}{{20}{c}}
{a - b} \\
{d - e} \\
{g - h}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1 \\
1 \\
{ - 1}
\end{array}} \right] \\
a - b = 1,d - e = 1,g - h = - 1. \\

Then, $$a - \left( { - 1} \right) = 1 \Rightarrow a = 0$$

d - \left( 2 \right) = 1 \Rightarrow d = 3 \\
g - \left( 3 \right) = - 1 \Rightarrow g = 2 \\

Also, M satisfies, $$M\left[ {\begin{array}{*{20}{c}} 1 \\\ 1 \\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\\ 0 \\\ {12} \end{array}} \right].$$

\left[ {\begin{array}{{20}{c}}
a&b;&c; \\
d&e;&f; \\
g&h;&i;
\end{array}} \right]\left[ {\begin{array}{{20}{c}}
1 \\
1 \\
1
\end{array}} \right] = \left[ {\begin{array}{{20}{c}}
0 \\
0 \\
{12}
\end{array}} \right]. \\
\left[ {\begin{array}{{20}{c}}
{a + b + c} \\
{d + e + f} \\
{g + h + i}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
0 \\
0 \\
{12}
\end{array}} \right] \\
a + b + c = 0,d + e + f = 0,g + h + i = 12. \\

Then, $(0) + ( - 1) + c = 0 \Rightarrow c = 1$ $ (3) + (2) + f = 0 \Rightarrow f = - 5 \\\ (2) + (3) + i = 12 \Rightarrow i = 7 \\\ $ So, we get, $$a = 0,{\text{ }}b = - 1,{\text{ }}c = 1,{\text{ }}d = 3,{\text{ }}e = 2,{\text{ }}f = - 5,{\text{ }}g = 2,{\text{ }}h = 3$$ and $$i = 7.$$ Thus, matrix, $M = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&1 \\\ 3&2&{ - 5} \\\ 2&3&7 \end{array}} \right]$ Now, we have been asked in the question to find the sum of the diagonal entries of M. So, the diagonal entries of M is, a, e and i. So, $$a + e + i = 0 + 2 + 7 = 9.$$ Thus, sum of the diagonal entries of M is $$9.$$ **So, the correct answer is “9”.** **Note** : Students should note that in the question we have only taken the value of one of the diagonals of a matrix, but we know that there are two diagonals, so another answer would have been equals to $$5,$$ since, $$c + e + g = 1 + 2 + 2 = 5,$$ is the sum of the entries of another diagonal.