Q: Prove that if and A and B are of the same size, then there exists a matrix C such that and A=CB.
A: Since we know and exists. So,
Now, let
Since and
Does that work? I guess A=I, B=I, and C=I. Do I need to show more?
I presume from the thread title that is the determinant of the matrix . I'm afraid your arguments only allow you to say that if are matrices with and , then which is vacuously true. That does not imply . Example: .
You started at the right place: Since are nonzero, they are invertible. Then , so if we can show , then we have found our !
Now we need some properties of the determinant, namely that and . Then it is easy to see that as required.