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Question
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
Solution
\[ ∆ = \begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
\[ = \begin{vmatrix}0 & 2 & 7 \\ 0 & 3 & 5 \\ 0 & 4 & 3\end{vmatrix} \left[ \text{ Applying } C_1 \to C_1 - 4 C_2 \right]\]
\[ \Rightarrow ∆ = 0\]
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