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

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.
• $\Delta=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{33}-a_{23}\times a_{31})+a_{13}(a_{21}\times a+{32}=a_{22}\times a_{31})$
Step 1:
$\Delta=\begin{bmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{bmatrix}$
Let us take 3x which is the common factor of $R_3$,
$\Delta=\begin{bmatrix}2 & 3 & 4\\5 & 6 & 8\\2 & 3 & 4\end{bmatrix}$
Step 2:
Since two rows are identical the value of the determinant is zero.
Hence $\Delta=0.$