(Multiplication by) a matrix is a LINEAR map-one consequence of this is it maps subspaces of a vector space to subspaces of another vector space.
If you have an $m \times n$ matrix with real entries, the vector spaces in question are $\Bbb R^n$ and $\Bbb R^m$, that is to say an $m \times n$ matrix when multiplied by "on the left" takes an $n$-tuple to an $m$-tuple:
$(m \times n)(n \times 1) \to m\times 1$.
So let's say you have a 3x2 matrix. Even though you are mapping INTO a space of a larger dimension, you don't wind up with the whole space "being reached", you only get (at best) a two-dimensional subspace...a plane. Which plane? The plane spanned by the two columns of our matrix:
$\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} =\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{bmatrix}\left(x\begin{bmatrix}1\\0\end{bmatrix}+y\begin{bmatrix}0\\1\end{bmatrix}\right)$
$= x\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} + y\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix}$
$ = x\begin{bmatrix}a_{11}\\a_{21}\\a_{31}\end{bmatrix} + y\begin{bmatrix}a_{12}\\a_{22}\\a_{32}\end{bmatrix}$.
Now if you start with a 2x3 matrix, you take a 3-dimensional space and shrink it to a (at most) 2-dimensional space. As you can see from the above, hitting the two-dimensional space with a 3x2 matrix doesn't "blow it back up" again, it will stay at most "two-dimensional".
So the net effect is this (at best, more shrinkage could occur if we have "bad matrices"), just looking at dimensionality:
$3 \to 2 \to 2$.
We lose an entire dimension, which is to say our "combo mapping" (the composition, or matrix product) isn't 1-1, it's "many-to-one". Such a mapping cannot possibly be invertible (any inverse isn't "uniquely defined", and some triples in $\Bbb R^3$ will "never be reached", so we can't find any "pre-image").
You should have much better luck if you try "the other product": a 2x3 times a 3x2 matrix, which does this to the dimensions (if your matrices are "nice"-that is, of maximum rank):
$2 \to 2 \to 2$.