1. Properties of 'special' matrices: Show the following:
(a) the product of two orthogonal matrices is orthogonal
(b) if A and B are hermitian, AB is only hermitian if [A, B] = 0
(c) if A and B are symmetric, [A, B] is antisymmetric
(d) if S is symmetric and A is antisymmetric, show that tr(SA) = 0
(e) the determinant of a unitary matrix is a complex number of magnitude 1
(f) if H is hermitian and U is unitary, show that U-1HU is hermitian
(g) if A is real and antisymmetric, show that (1 - A)(1 + A)-1 is orthogonal
(h) if H is a hermitian matrix, show that eiH is unitary
(i) that [M, ABC] = [M, A]BC + A[M, B]C + AB[M, C]