Let A = [ a _ ij ] be a 3 3 matrix such that A [ l 0 1 0 ]= [ l 0… - Vidyadip

MCQ

Let $A =\left[ a _{ ij }\right]$ be a $3 \times 3$ matrix such that $A \left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A \left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$ and $A \left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$, then $a _{23}$ equals:

✓
-1
B
0
C
2
D
1

✓

Answer

Correct option: A.

-1

(A)
Let $A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$
$A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right] \Rightarrow\left[\begin{array}{l}a_{12} \\ a_{22} \\ a_{32}\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \Rightarrow \begin{array}{l}a_{22}=0 ; a_{12}=0 \\ a_{32}=1\end{array}$
$A \left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \Rightarrow \begin{array}{l}4 a _{11}+ a _{12}+3 a _{13}=0 \\ 4 a _{21}+ a _{22}+3 a _{23}=1 \\ 4 a _{31}+ a _{32}+3 a _{33}=0\end{array} \Rightarrow 4 a _{21}+3 a _{23}=1$
$\begin{array}{l} A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right] \Rightarrow \begin{array}{l}2 a_{11}+a_{12}+2 a_{13}=1 \\ 2 a_{21}+a_{22}+2 a_{23}=0 \\ 2 a_{31}+a_{32}+2 a_{33}=0\end{array} \Rightarrow a_{21}+a_{23}=0 \\ -4 a_{23}+3 a_{23}=1 \Rightarrow a_{23}=-1\end{array}$

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