I will tell bit faster method to solve such questions
System is
$\begin{bmatrix} 1 & 2&3 \\ 1 & 3 &4 \\ 2& 2 & 3 \end{bmatrix}$ $\begin{bmatrix} x\\ y\\ z \end{bmatrix}$ = $\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$
So, what question asks is what combination of (X*column1) + (Y*column2) +(Z*column3) produces the vector on the right-hand side.
$X\begin{bmatrix} 1\\ 1\\ 2 \end{bmatrix}$ + $Y\begin{bmatrix} 2\\ 3\\ 2 \end{bmatrix}$ + $Z\begin{bmatrix} 3\\ 4\\ 3 \end{bmatrix}$ =$\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$
(A)X=6, Y=3, Z=2
LHS comes to be while evaluating
$\begin{bmatrix} {\color{Red} 6} &{\color{Red} 6} &{\color{Red} 6} \\ & & \\ & & \end{bmatrix}$
The first column will not evaluate to be 6 on RHS(6+6+6 is not equal to 6), so we stop here only and reject option (A).
Option (B) X=12, Y=3, Z=-4
$\begin{bmatrix} {\color{Green} 12} & {\color{Green} 6} & {\color{Green} -12}\\ {\color{Red} 12} &{\color{Red} 9} & {\color{Red} -16} \\ & & \end{bmatrix}$
The second entry of column2 does not add up to match 8 on the RHS so we stop our calculation here and reject this option too.
Option (c) X=6, Y=6, Z=-4
$\begin{bmatrix} 6 & 12 &-12 \\ 6 & 18 &-16 \\ 12 &12 & -12 \end{bmatrix}$ = $\begin{bmatrix} 6\\ 8\\ 12 \end{bmatrix}$
Yes this matches our answer. Ans-(C)
At first this method may look complicated but this is fastest method to solve such question.Not more than 50Secs. :P
