Evaluate the following:

Let $\Delta=\left|\begin{array}{lll}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|$

$\Delta=\left|\begin{array}{lll}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|$

$=\left|\begin{array}{ccc}x-1 & 1-x & 0 \\ 1 & x & 1 \\ 0 & 1-x & x-1\end{array}\right| \quad\left[\right.$ Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $\left.R_{3} \rightarrow R_{3}-R_{2}\right]$

$=(x-1)^{2}\left|\begin{array}{ccc}1 & -1 & 0 \\ 1 & x & 1 \\ 0 & -1 & 1\end{array}\right|$

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

$=(x-1)^{2}(x+1+1)$           [Expanding along last row]

$=(x-1)^{2}(x+2)$

$\therefore \Delta=(x-1)^{2}(x+2)$