Evaluate :

$\Delta=\left|\begin{array}{ccc}x+\lambda & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda\end{array}\right|$

$=\left|\begin{array}{rrr}\lambda & 0 & x \\ -\lambda & \lambda & x \\ 0 & -\lambda & x+\lambda\end{array}\right| \quad\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}-C_{2}, C_{2} \rightarrow C_{2}-C_{3}\right]$

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

$=\lambda\left|\begin{array}{cc}0 & 2 x+\lambda \\ -\lambda & x+\lambda\end{array}\right|+x\left|\begin{array}{cc}-\lambda & 0 \\ 0 & -\lambda\end{array}\right|$

$=\lambda[\lambda(2 x+\lambda)]+x \lambda^{2}$

$=\lambda^{2}\left(2 x+\lambda+\lambda^{2} x\right)$

$=3 \lambda^{2} x+\lambda^{3}$

$=\lambda^{2}(3 x+\lambda)$