Hello, I just wanted to make sure my proof is correct before turning it in. It's a very basic proof, but it's been a while since I have proved anything.
Q: Prove that if the product AB is a square matrix, then the product BA is defined.
A: Suppose the product is not a square matrix. Then, by the definition of matrix multiplication, must be an n matrix and an p matrix where (by out supposition). It follows that is not defined, because the column of matrix is not equal to row of matrix . But, our original statement reads is defined. Therefore, by contradiction we have proven that if the product is a square matrix, then the product is defined.
The only way for the the product to exist is for to be a and for to be a .
If the product is square what does that tell us about ?
So, I should just go with a direct proof.