To be honest, I have no idea what you are talking about, in the statement and your "proof". What do you mean by ? Are supposed to be matrices? If so, your formula is false, in general. You really want . Also, what are and ?
So, my problem is, how to proof this is true?
What I could do:
Lets Say and , so, in this case, , and obviously . The only way of getting is multiplying for , which would be false if we used , since . Even though this makes sense, is this a valid proof? And is there a cleaner one?
PS: A and B are matrices.
Let be the ith row of .
Let be the jth column of .
So the i,jth entry of the product
Now lets compare with
Now this exchanges the rows and columns of A and B
So
Now what can we conclude?