PLEASE HELP!!!

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

THANKS!