Solve the equation

Given, $\quad\left|\begin{array}{ccc}y+z & z & y \\ z & z+x & x \\ y & x & x+y\end{array}\right|$

[Applying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$ ]

$=\left|\begin{array}{ccc}2(y+z) & z & y \\ 2(z+x) & z+x & x \\ 2(y+x) & x & x+y\end{array}\right|$

$=2\left|\begin{array}{ccc}y+z & z & y \\ z+x & z+x & x \\ x+y & x & \dot{x}+y\end{array}\right|$

Now,

[Applying $C_{1} \rightarrow C_{1}-C_{2}$ ]

$=2\left|\begin{array}{ccc}y & z & y \\ 0 & z+x & x \\ y & x & x+y\end{array}\right|$

Next,

[Applying $C_{3} \rightarrow C_{3}-C_{1}$ ]

$=2\left|\begin{array}{ccc}y & z & 0 \\ 0 & z+x & x \\ y & x & x\end{array}\right|$

Lastly,

[Applying $R_{3} \rightarrow R_{3}-R_{1}$ ]

$=2\left|\begin{array}{ccc}y & z & 0 \\ 0 & z+x & x \\ 0 & x-z & x\end{array}\right|$

$=2 y[(z+x) x-x(x-z)]=2 y[2 x z]=4 x y z$