linear algebra - T/F: If $A$ and $B$ are matrices such that AB=$I_n$, then $A$ and $B$ are square matrices.

In my book (Linear Algebra by Steven Andrilli 4th ed.) There is a true/false question in the chapter 2 review that's all about "systems of linear equations." The question asks:

T/F: If $A$ and $B$ are matrices such that AB=$I_n$, then $A$ and $B$ are square matrices.

I said that this was true. The correct answer according to the book is False.

If $A$ is $m$x$n$ non-square matrix and $B$ is $n$x$p$ square matrix, then $A$*$B$ is an $m$x$p$ matrix. I don't understand how it can be square if $A$ or $B$ is non square.