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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics2 Marks
MCQ
Let M be a $3 \times 3$ matrix satisfying $M \left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right], M \left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]$ 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
  • A
    7
  • B
    8
  • ✓
    9
  • D
    6
✓

Answer

Correct option: C.
9
(C) Let $M =\left[\begin{array}{lll} a & b & c \\ x & y & z \\ l & m & n \end{array}\right]$, then
$M \left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right] \Rightarrow\left[\begin{array}{l} b \\ y \\ m\end{array}\right]=\left[\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right]$
∴ by the equality of matrices,
$\begin{array}{l} b =-1, y=2, m=3 \\ M \left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right] \Rightarrow\left[\begin{array}{l} a - b \\ x-y \\ l- m \end{array}\right]=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]\end{array}$
∴ by the equality of matrices,
$\begin{array}{l} a - b =1, x-y=1, l- m =-1 \\ \Rightarrow a =0, x=3, l=2\end{array}$
$M \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ 0 \\ 12\end{array}\right] \Rightarrow\left[\begin{array}{l} a + b + c \\ x+y+ z \\ l+ m + n \end{array}\right]=\left[\begin{array}{c}0 \\ 0 \\ 12\end{array}\right]$
∴ by the equality of matrices,
$a + b + c =0, x+y+ z =0, l+ m + n =12$
$\Rightarrow c=1, z=-5, n=7$
∴ sum of diagonal elements of $M = a +y+ n$
$=0+2+7=9$

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