Question

Write the value of the determinant \[\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}\]

Answer 1

\[\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}\]
\[ = \begin{vmatrix}x + y + z & x + y + z & z + x + y \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix} \left[\text{ Applying }R_1 \to R_1 + R_2 \right]\]
\[ = \left( x + y + z \right)\begin{vmatrix}1 & 1 & 1 \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix} \left[\text{ Taking }\left( x + y + z \right)\text{ common from }R_1 \right]\]
\[ = \left( x + y + z \right)\begin{vmatrix}1 & 1 & 1 \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 + 3 R_1 \right]\]
\[ = \left( x + y + z \right)\begin{vmatrix}1 & 1 & 1 \\ z & x & y \\ 0 & 0 & 0\end{vmatrix}\]
\[ = 0 \left[\text{ Expanding along the last row }\right]\]
Hence, the value of the determinant 

\[\begin{vmatrix}x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3\end{vmatrix}\] is 0.