Question

Let $\lambda$ and $\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations $\lambda x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0,x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0$ and $-x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0$ Has a non-trivial solution. For $\lambda =1$, find all values of $\alpha$

Answer 1

For non-trivial solution

$\left|\begin{array}{ccc}\lambda & \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \\ -1& \sin \alpha & -\cos \alpha \end{array}\right|=0$

$\lambda (-{\cos }^2\alpha -{\sin }^2\alpha )-\sin \alpha (\sin \alpha -\cos \alpha )+\cos \alpha (\sin \alpha +\cos \alpha )=0$.

$-\lambda ({\sin }^2\alpha +{\cos }^2\alpha )-{\sin }^2\alpha +{\cos }^2\alpha -\sin \alpha \cos \alpha +\sin \alpha \cos \alpha =0.$

$\lambda ={\cos }^2\alpha -{\sin }^2\alpha$

$\lambda =\cos 2\alpha$

$\lambda$ can be form $[-1,1]$ because range of $\cos 2\alpha$ is form $[-1,1]$.

For $\lambda =1$

$\cos 2\alpha =1$

$\alpha =n\pi$. where $n$ is an integer.