Prove that

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

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

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

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

[Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$ ]

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

$=(a+x+y+z)\left|\begin{array}{ll}a & 0 \\ 0 & a\end{array}\right|=a^{2}(a+z+x+y)$