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Question
Find the value of the determinant \[\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .\]
Solution
\[\begin{vmatrix} 243 & 156 & 300\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix}\]
\[ = \begin{vmatrix} 243 - \left( 81 \times 3 \right) & 156 - \left( 52 \times 3 \right) & 300 - \left( 100 \times 3 \right)\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 - 3 R_2 \right]\]
\[ = \begin{vmatrix} 0 & 0 & 0\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix} \]
\[ = 0\]
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