Question

Let $\lambda$ and $\alpha$ be real. Find the set of all value of $\lambda$ for which the system of linear equations $\lambda x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0,x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0,-x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0$.

Answer 1

$\begin{array}{c}\left(\begin{array}{ccc}\lambda & \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \\ -1& \sin \alpha & -\cos \alpha \end{array}\right)=0\\ \Rightarrow -\lambda \left|{\cos }^2\alpha +{\sin }^2\alpha \right|-\sin \alpha \left(\sin \alpha -\cos \alpha \right)+\cos \alpha \left(\sin \alpha +coa\alpha \right)\\ \Rightarrow -\lambda -{\sin }^2\alpha +\sin \alpha \cdot \cos \alpha +{\cos }^2\alpha +\sin \alpha \cdot \cos \alpha =0\\ \Rightarrow -\lambda +\cos \alpha +{\sin }^2\alpha =0\\ \Rightarrow \lambda =\sqrt{2}\sin \left(2\alpha +\frac{\pi }{4}\right),if\lambda =\sqrt{2},\frac{\pi }{2}=2\alpha +\frac{\pi }{4}\\ \Rightarrow \alpha =\frac{\pi }{8},if\lambda =1,\frac{\pi }{4}=2\alpha +\frac{\pi }{4},\alpha =0,if\lambda =-1,-\sqrt{2},\alpha =\frac{-3\pi }{8},-\frac{\pi }{4}\end{array}$