The system of equations $\lambda x+y+z=0,-x+\lambda y+z=0,-x-y+\lambda z=0$ will have a non- trivial solution for  real values of $\lambda$are

$The condition for homogeneous system to have a non-trivial solution is \Delta =0$

$\begin{array}{l}\Rightarrow \left|\begin{array}{ccc}\lambda & 1& 1\\ -1& \lambda & 1\\ -1& -1& \lambda \end{array}\right|=0\\ \Rightarrow \lambda \left[{\lambda }^2+1\right]-[-\lambda +1]+[1+\lambda ]=0\\ \Rightarrow {\lambda }^3+\lambda +\lambda -1+1+\lambda =0\Rightarrow {\lambda }^3+3\lambda =0\\ \Rightarrow \lambda \left({\lambda }^2+3\right)=0\\ \Rightarrow \lambda =0\end{array}$