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Question
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
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Solution
\[A = \begin{vmatrix} 2 & - 3 & 5\\4 & - 6 & 10\\6 & - 9 & 15 \end{vmatrix}\]
\[ = \begin{vmatrix} 2 & - 3 & 5\\4 - 4 & - 6 + 6 & 10 - 10\\6 & - 9 & 15 \end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - 2R {}_1 \right]\]
\[ = \begin{vmatrix} 2 & - 3 & 5\\0 & 0 & 0\\6 & - 9 & 15 \end{vmatrix} \]
\[ = 0\]
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