The system of linear equations $x+\lambda y-z=0,\lambda x-y-z=0;x+y-\lambda z=0$has a non-trivial solution , then sum of cubes of $\lambda$is

$System has non-trivial solution\Rightarrow \Delta =0$

 $$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}1& \lambda & -1\\ \lambda & -1& -1\\ 1& 1& -\lambda \end{array}\right|=0 \\ \Rightarrow 1\left(\lambda +1\right)-\lambda \left(-{\lambda }^2+1\right)-1\left(\lambda +1\right)=0 \\ \Rightarrow \lambda \left({\lambda }^2-1\right)=0 \\ \Rightarrow \lambda =-1,0,1 \\ \therefore sum of cubes of \lambda =-1+0+1=0 \end{gathered}$$