Let M be a 3 3 matrix satisfying M [ l 0 1 0 ]= [ c -1 2 3 ], M [… - Vidyadip

MCQ

Let $M$ be a $3 \times 3$ matrix satisfying $M\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right], \quad M\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]$, and $M\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ 0 \\ 12\end{array}\right]$ Then the sum of the diagonal entries of $M$ is

✓
$9$
B
$8$
C
$7$
D
$6$

✓

Answer

Correct option: A.

$9$

a
Let M be $\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$.

Now from the first condition $\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]\left[\begin{array}{l}0 \\ 1 \\ o \end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right]$

we get the values of $b, e, h$ as $-1,2,3$ respectively.

Newly formed matrix $M=\left[\begin{array}{ccc}a & -1 & c \\ d & 2 & f \\ g & 3 & i\end{array}\right]$

From the second condition

$\left[\begin{array}{ccc} a & -1 & c \\ d & 2 & f \\ g & 3 & i \end{array}\right]\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]$

Solving the matrix, we get three equations

$\Rightarrow a+1=1, d-2=1, g-3=-1$

$\Rightarrow a=o, d=3, g=2$

Modified matrix $M =\left[\begin{array}{ccc} o & -1 & c \\ 3 & 2 & f \\ 2 & 3 & i \end{array}\right]$

Now the third and the last condition, $\left[\begin{array}{ccc}0 & -1 & c \\ 3 & 2 & f \\ 2 & 3 & i \end{array}\right]\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ o \\ 12\end{array}\right]$

$\Rightarrow 2+3+ i =12 \Rightarrow i =7$

Sum of the diagonal entries of $M = a + e + i = o +2+7=9$

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