The value of \( \begin{vmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{vmatrix} \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0
The third row is 3x times the first row:
\( \displaystyle (6x,\ 9x,\ 12x)=3x\,(2,\ 3,\ 4) \)
Taking 3x common from the third row:
\( \displaystyle \begin{vmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{vmatrix}=3x\begin{vmatrix}2 & 3 & 4\\5 & 6 & 8\\2 & 3 & 4\end{vmatrix} \)
The last determinant has two identical rows (R1 = R3), so its value is 0. Hence the given determinant is 0 for every x.