# Math Help - Simple proof

1. ## Simple proof

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 $AB$ is not a square matrix. Then, by the definition of matrix multiplication, $A$ must be an $m\times$n matrix and $B$ an $n\times$p matrix where $m\neq{p}$(by out supposition). It follows that $BA$is not defined, because the column $p$ of matrix $B$ is not equal to row $m$ of matrix $A$. But, our original statement reads $BA$is defined. Therefore, by contradiction we have proven that if the product $AB$is a square matrix, then the product $BA$ is defined.

Thanks

2. The only way for the the product $AB$ to exist is for $A$ to be a $j \times m$ and for $B$ to be a $m \times n$.
If the product is square what does that tell us about $j~\&~n$?

3. $j=n=m$

...right?

So, I should just go with a direct proof.

4. Originally Posted by Danneedshelp
$j=n=m$.
It does mean tha $j=n$. But not necessarily $j=n\color{red}=m$
What about a $m \times n$ times a $j \times m$?