Write the value of the determinant
Question:

Write the value of the determinant $\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$.

Solution:

$\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$

$=\left|\begin{array}{ccc}x+y+z & x+y+z & z+x+y \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$   $\left[\right.$ Applying $\left.R_{1} \rightarrow R_{1}+R_{2}\right]$

$=(x+y+z)\left|\begin{array}{ccc}1 & 1 & 1 \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$                 [Taking $(x+y+z)$ common from $R_{1}$ ]

$=(x+y+z)\left|\begin{array}{ccc}1 & 1 & 1 \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$            $\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}+3 R_{1}\right]$

$=(x+y+z)\left|\begin{array}{lll}1 & 1 & 1 \\ z & x & y \\ 0 & 0 & 0\end{array}\right|$

$=0 \quad$ [Expanding along the last row]

Hence, the value of the determinant $\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ -3 & -3 & -3\end{array}\right|$ is 0 .

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