Prove the following for matrices \(A, B\), and \(C\).

a. \((A B) C=A(B C)\).

b. \((A B)^{T}=B^{T} A^{T}\)

c. \(\operatorname{tr}(A)\) is invariant under similarity transformations.

d. If \(A\) and \(B\) are orthogonal, then \(A B\) is orthogonal.