Question

Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]

Answer 1

\[A = \begin{vmatrix} 2 & - 3 & 5\\4 & - 6 & 10\\6 & - 9 & 15 \end{vmatrix}\] 
\[ = \begin{vmatrix} 2 & - 3 & 5\\4 - 4 & - 6 + 6 & 10 - 10\\6 & - 9 & 15 \end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - 2R {}_1 \right]\] 
\[ = \begin{vmatrix} 2 & - 3 & 5\\0 & 0 & 0\\6 & - 9 & 15 \end{vmatrix} \] 
\[ = 0\]