what are the eigenvalues in orthgonal matrix, How to explain? what are the possible eigenvalues of an orthogonal matrix?
I got the answer key which says its 1 and -1 but it doesn't explain well

Here is an approach. Recalling the fact 

if $\lambda$ is the eigenvalue of $A$, then $\lambda^{-1}$ is the eigenvalue of $A^{-1}$,

we have
$$ \langle Ax,x \rangle =  \langle x, A^T x \rangle = \langle x, A^{-1} x \rangle $$
$$ \implies \langle \lambda x,x \rangle  = \langle x, \frac{1}{\lambda} x \rangle $$ 
$$ \implies \lambda \langle  x,x \rangle  =\frac{1}{\bar \lambda} \langle x,  x \rangle $$ 
$$ |\lambda|^2 = 1 .$$
