I'm not sure how you went from line 2 to line 3, sparky. It looks like you're assuming that commutes with and , which you haven't proved (and which, without thinking too much about it, I doubt is true).
The facts that and the 's are diagonal both play key roles in the proof. Can you show that as topsquark suggests?
Thanks for your reply LoblawsLawBlog.
I think I now understand. Correct me if I'm wrong:
Since and I is an identity matrix, therefore I is a diagonal matrix. C is also equal to I, therefore C is also diagonal. And since and are also diagonal, therefore since all of them are diagonal, AB = BA
same thing: and, since and are diagonal matrices they commute: this is the same as that you had before.
If this is correct, then what was the purpose of ? To confuse me? What about the being diagonal matrices - was that designed to confuse me as well? Or did they have some real purpose in this question?