Prove the following identities:

LHS :

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

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

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

$=-2 x\left|\begin{array}{ccc}0 & 1 & 1 \\ z & z+x & x \\ y & x & x+y\end{array}\right| \quad\left[\right.$ Taking $-2 x$ common from $\left.R_{1}\right]$

$=-2 x\left|\begin{array}{ccc}0 & 0 & 1 \\ z & z & x \\ y & -y & x+y\end{array}\right| \quad$ [Applying $C_{2} \rightarrow C_{2}-C_{3}$ ]

$=-2 x(-z y-z y) \quad$ [Expanding along first row]

$=4 x y z$

$=$ RHS

$\therefore\left|\begin{array}{ccc}y+z & z & y \\ z & z+x & x \\ y & x & x+y\end{array}\right|=4 x y z$