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 for

$The system has non-trivial solution \Rightarrow \Delta =0$

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

$\Rightarrow \lambda ({\lambda }^2-1)=0$

$\Rightarrow \lambda =0,\pm 1$

$\therefore \lambda$ has three values