Problem 2 Suppose n x n matrices A, B are similar (i.e. there is an invertible matrix S such that S-AS = B). i) Let A be a scalar. By directly expanding (using distributivity of matrix multiplication), show that S-'(A - XIn)S = B - XIn. ii) Using the multiplicativity of the determinant and the fact that det (S ) = (det S)-, show that det(A - XIn) = det(B - XIn). iii) Conclude that fA(X) = fB(X). Consequently, show that tr (A) = tr (B), and that the eigen- values of A and B are the same