Let $\lambda$and $\alpha$ be real. The set of all values of for which the system of linear equations$\lambda x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0$,$x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0$,$-x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0$has a non-trival solution is

$$\begin{gathered} The condition for a homogeneour system of equations to have a non-trivial solution is \Delta =0 \\ \Rightarrow \Delta =\left|\begin{array}{ccc}\lambda & \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \\ -1& \sin \alpha & -\cos \alpha \end{array}\right|=0 \\ \Rightarrow \lambda \left(-{\cos }^2\alpha -{\sin }^2\alpha \right)-\sin \alpha \left(-\cos \alpha +\sin \alpha \right)+\cos \alpha \left(\sin \alpha +\cos \alpha \right)=0 \\ \Rightarrow \lambda =\cos 2\alpha +\sin 2\alpha \\ The range of \cos 2\alpha +\sin 2\alpha is \left[-\sqrt{2},\sqrt{2}\right]. \\ \therefore The range of \lambda is \left[-\sqrt{2},\sqrt{2}\right] \end{gathered}$$