linear algebra - AB = I BA = I, proof not valid?

Ok so I'm right at the end of the first chapter in my book which means I just know some matrix/vector operations, what it means for a matrix to be invertible and what the transpose of a matrix is. Add to that some of the basic rules regarding the inverse and transpose of a matrix, such as $(AB)^{-1} = B^{-1}A^{-1}$

One questions asks me to prove
$AB = I \iff BA = I$
I googled but all I could find was proofs that used concepts I've never heard of, like determinants. The reason I'm asking for help is because I think the book's answer is wrong. Here it is for $AB = I \implies BA = I$:

$B = BI$
$B(AB) = (BA)B$
Post-multiply this $B = (BA)B$ by $B^{-1}$:
$BB^{-1} = (BA)BB^{-1}$
$I = (BA)I = BA$
We have $BA = I$

He proves $BA = I \implies AB = I$ in a similar way. The thing I am turning against is that it is assumed that $B$ is invertible, that $B^{-1}$ exists. The question never said anything about $B$ being invertible. But if $B^{-1}$ actually exists then no proof is needed since then by the definition of an invertible matrix, $A$ is the inverse of $B$ and $AB = BA = I$. So am I wrong or is the proof in the book not really correct? Or is it a valid assumption that $B$ is invertible?

Answer

There are thre options here: either you are sloppy or the book is or both.

It is pretty easy to show the following fact:

If $A$ and $B$ are square matrices, then $AB$ is invertible if and only if $A$ and $B$ are invertible.

Using this fact, the proof used in your book is "correct".

1. If the statement above was already proven in your book, then it may be the case that you missed that fact (so you are sloppy)


2. If the statement was not proven in the book, then the book is sloppy.


3. If the statement was proven, but the book did not explain well enough that it was used in the proof you cite, then it may be that both of you are sloppy.