Can functions within a matrix adjust its size? I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix:
\begin{bmatrix}G\begin{pmatrix}1\\0\\0\end{pmatrix}&G\begin{pmatrix}0\\1\\0\end{pmatrix}&G\begin{pmatrix}0\\0\\1\end{pmatrix}\end{bmatrix}
could become a n by 3 matrix, for any whole number n (for example in the case above I would want to resolve it to a 5 by 3 matrix.
Is this possible? Or is it back to the drawing board? 

That is mostly a question of notation and convention. You can certainly choose to define that in your work, the notation
$$ A = \begin{bmatrix}X & Y & Z\end{bmatrix} $$
where $X$, $Y$ and $Z$ are column matrices of the same height, will mean that $A$ is the $3\times n$ matrix that has those three columns. As a matter of terminology that would mean that you're defining $A$ as a block matrix.
However, it seems unlikely that this is actually what your problem depends on. There's essentially no mathematical content in asking (effectively) "is it possible to form block matrices" -- the answer to that will only tell you how your notation works, but not anything about the underlying mathematical structure that your notation speaks about.
So when you say that your problem "comes down to" whether you can employ that notation or not, it sounds very likely that you have some conceptual confusion or mistake in your work before you reach that point. And you should probably ask another question where you give details of that work and ask whether your approach is legitimate.
