Question: Can anyone prove that a [Weyl Algebra](http://en.wikipedia.org/wiki/Weyl_algebra) is not isomorphic to a matrix ring over a division ring?

Answer: 
I would only like to add a simple proof that the Weyl algebra doesn't even HAVE any (non-trivial) finite-dimensional representations. Already in the case n=1, consider the relations 
$$[\partial\_x,x]=1.$$
Now suppose you had a finite dimensional representation, and take the trace of both sides of the above.

It implies that the identity acts as 0 so the whole representation does.
A matrix ring contains zero-divisors.