Let  $\Delta =\left|\begin{array}{ccc}\lambda & 1& 1\\ 1& \lambda & 1\\ 1& 1& \lambda \end{array}\right|=\left|\begin{array}{ccc}\lambda +2& 1& 1\\ \lambda +2& \lambda & 1\\ \lambda +2& 1& \lambda \end{array}\right|\left[C_1\to C_1+C_2+C_3\right]$
$=(\lambda +2)|\begin{array}{ccc}1& 1& 1\\ 1& \lambda & 1\\ 1& 1& \lambda \end{array}=|\left|\begin{array}{ccc}1& 0& 0\\ 1& \lambda -1& 0\\ 1& 0& \lambda -1\end{array}\right|$ 
[ using $C_2\to C_2-C_1$  and $C_3\to C_2-C_1$  ] =  $(\lambda +2){(\lambda -1)}^2$
If $\Delta =0$  , then $\lambda =-2$  or  $\lambda =1$. 
But when  $\lambda =1$, the system of equation becomes $x_1+x_2+x_3=1$ which has infinite number of solutions. When $\lambda =-2$  , by adding three equations, we obtain 0 = 3 and thus, the system of equations is inconsistent.