a) Let A=
1 0 0
0 2 0
0 0 3
Suppose that B is a 3x3 matrix and that AB=BA. Show that B is also a diagonal 3x3 Matrix
b) Suppose that A is an 3x3 real matrix with three distinct eigenvalues. Let B be another 3x3 matrix for which AB=BA. Show that B is diagonalizable.
c) Suppose that A and B are diagonalizable matrices with precisely the same eigenspaces (but not necessarily the same eigenvalues). Prove that AB=BA