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: 
Notation: The Weyl algebra is 
$$k[x\_1, x\_2, \ldots, x\_n, \partial\_1, \partial\_2, \ldots, \partial\_n]$$
with the obvious relations.

The Weyl algebra doesn't contain any division rings larger than $k$, and it is infinite dimensional over $k$. So, assuming you don't allow infinite matrices, that's a proof.

How to see that it doesn't contain any division ring larger than $k$? I just need to show that any nonconstant differential operator is not invertible. One way to see this is to notice that multiplying differential operators multiplies symbols, and the symbol of a nonconstant differential operator is a nonconstant polynomial.
A matrix ring contains zero-divisors.