[SOLVED] Linear Algebra- Diagonalizability

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!

Quote:

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Originally Posted by kutcha

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

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If you suppose B to be an arbitrary matrix, say
B =
a b c
d e f
g h i

and perform the matrix multiplication on both sides, you should come up and be forced that the values of b,c,d,f,g,h is 0..