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**hashshashin715** first one is if A and B are unitary, then prove (or disprove) A + B is unitary.

Both $\displaystyle I\,,\,-I$ 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.