Prove the following identities:

LHS

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

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

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

$=(a+x+y+z)\left|\begin{array}{lll}1 & y & z \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right|$            [Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$ ]

$=(a+x+y+z) a^{2} \quad$ [Expanding along first column]

$=a^{2}(a+x+y+z)$

$=$ RHS

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