first one is if A and B are unitary, then prove (or disprove) A + B is unitary.
Both are unitary, so...
so we have AA* = I and BB* = I.
AA* + BB* = 2I
Can you do this: (A + B)* = (A* + B*)?
If you can, then we can have (A + B)(A + B)* = AA* + AB* + BA* + BB* = I
so we would only need to prove AB* + BA* = -I. How do you do that?
2nd proof:We have a real orthogonal n x n matrix A. If lamda is a complex eigenvalue of A, then prove the complex conjugate of lamda is also an eigenvalue of A. In other words, the non real eigenvalues of A occur in conjugate pairs.