Question:
Write the value of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 4 & -6 & 10 \\ 6 & -9 & 15\end{array}\right|$.
Solution:
$=\mid 2 \quad-3 \quad 5$
$4-4-6+6 \quad 10-10$
$\begin{array}{lll}6 & -9 & 15 \mid\end{array}$ [Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}$ ]
$=\mid \begin{array}{lll}2 & -3 & 5\end{array}$
$\begin{array}{lll}0 & 0 & 0\end{array}$
$\begin{array}{|lll|}6 & -9 & 15\end{array}$
$=0$