does a rectangular matrix have an inverse? I know all square matrices have easily to identify inverses, but does that continue on with rectangular matrices?

Actually, not all square matrices have inverses.  Only the invertible ones do.  For example, $\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$ does not have an inverse.
And no, non-square matrices do not have inverses in the traditional sense.
There is the concept of a generalized inverse.  To very briefly summarize the link, an $n \times m$ matrix $A$ has an $m \times n$ generalized inverse, denoted $A^g$, if $A^g$ satisfies $A A^g A = A$.
