Question

Without expanding, show that the value of the following determinant is zero:

\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]

Answer 1

\[ ∆ = \begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
\[ = \begin{vmatrix}0 & - 3 & 2 \\ 0 & - 1 & 2 \\ 0 & 5 & 2\end{vmatrix} \left[ \text{Applying }C_1 \to C_1 + 2 C_2 \right]\]
\[ \Rightarrow ∆ = 0\]