Let $A$ be an orthogonal matrix, and let $λ$ be an eigenvalue of $A$. Prove $λ=1$ or $λ=-1$.
What I did was, as $A$ is orthogonal we know the following holds: $$\lVert Av\rVert = \lVert v\rVert \implies \left(Av,Av\right)=\left(v,v\right) \implies \frac{\left(Av,Av\right)}{\left(v,v\right)}=1.$$ Then, $$Av=λv \implies \lVert Av\rVert =\lVert λv\rVert \implies \left(Av,Av\right)=\left(λv,λv\right)\implies \frac{\left(Av,Av\right)}{\left(v,v\right)}=λ$$ which means $λ$ must equal $1$. What am I missing?
Edit for future readers:
The correct way to get $λ$ out of the inner product should have been
$$\left(λv,λv\right) \implies λ\left(v,λv\right)\implies λ\left(λv,v\right) \implies \ λ^{2}\left(v,v\right).$$