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Question
The value of the determinant \[\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}\] is
Options
9x2(x + y)
9y2(x + y)
3y2(x + y)
7x2(x + y)
Solution
\[\begin{vmatrix}x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x\end{vmatrix}\]
\[ = \begin{vmatrix}- 2y & y & y \\ x + 2y & x & x + y \\ - y & 2y & - y\end{vmatrix} \left[\text{ Applying }R_1 \to R_1 - R_2\text{ and }R_3 \to R_3 - R_2 \right]\]
\[ = y^2 \begin{vmatrix}- 2 & 1 & 1 \\ x + 2y & x & x + y \\ - 1 & 2 & - 1\end{vmatrix} \left[\text{ Taking }\left( y \right)\text{ common from }R_1\text{ and from }R_3 \right]\]
\[ = y^2 \begin{vmatrix}- 2 & - 3 & 3 \\ x + 2y & 3x + 4y & - y \\ - 1 & 0 & 0\end{vmatrix} \left[\text{ Applying }C_2 \to C_2 + 2 C_1\text{ and }C_3 \to C_3 - C_1 \right]\]
\[ = y^2 \left[ - 1\left( 3y - 9x - 12y \right) \right]\]
\[ = y^2 \left[ 9y + 9x \right]\]
\[ = 9 y^2 \left( y + x \right)\]
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