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Question
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
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Solution
\[ ∆ = \begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
\[ = \begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 13 & 17 & 5\end{vmatrix} \left[ \text{ Applying } R_3 \to R_3 - R_1 \right]\]
\[ \Rightarrow ∆ = 0\]
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