(a) All eigenvalues of A are $\pm 1$ whenever A is unitary and Hermetian. Prove the Hermetian part of this (they already took care of the unitary part).

(b) not sure...

(c) I saw the answer but was not able to figure out a key part of the proof. If we let $\mathbf{x} = \mathbf{v},$ we get $F\mathbf{v} = -\mathbf{v}.$ That means that exactly one of the eigenvalues is -1, but I am not sure why.