# Write the value of the determinants $\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}$

Toolbox:
• The Value of the determinant of a $3\times 3$ matrix can be obtained by$\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31}& a_{32} & a_{33}\end{bmatrix}$
• If two rows (or columns) are identical,the value of the determinant is 0.
• $\Delta =a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{33}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}-a_{22}\times a_{31})$
Given $\Delta=\begin{bmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{bmatrix}$

Let $\Delta=\begin{bmatrix}2 & 3 & 4\\5 & 6 & 8\\6x&9x &12x\end{bmatrix}$

Let us take 3x which is the common factor of $R_3$,then

$\Delta=\begin{bmatrix}2 & 3 & 4\\5 & 6 & 8\\2 & 3 & 4\end{bmatrix}$

Since two rows are identical ,the value of the determinant is zero.

Hence $\Delta=0.$

answered Mar 11, 2013