If yours matrix is "generic" (i.e. You do not suspect there are some specific algebraic relation between elements) then (as far as I know) there is nothing better than just use LU decomposition carefully putting elements on the right or on the left. (I mean LU can be easyly generalized to noncommutative case but formulas will be a little more complicated).

Actually the formulas which you write in 2*2 are very instructive !
The expressions A - B D^{-1} C called "Schur complements" see
http://en.wikipedia.org/wiki/Schur_complement

You can obtain inversion of non-commutative matrix in this way - just consider that D - is $(n-1) \times (n-1)$ matrix since you did not use commutativity in yours formulas - they will work in this case also.
So you can inductively go on till $n=1$ - so you obtain the inversion of yours matrix.
Actually this is the LU algorithm.

There some math involved for specific matrices with non-commutative entries.
For example if and only if [a,c] = [b,d] =0 and [a,d] = [b,d],
than you can see that the inverse matrix can be given by the same formula as in commutative
case

$$ \frac{ 1 }{ad-cb} d ~~~\frac{ -1 }{ad-cb}b $$
$$ \frac{ -1 }{ad-cb}c ~~~\frac{ 1 }{ad-cb}a $$

Similar fact holds true for $n\times n$ matrix if each 2 by 2 submatrix satisfy the relations above.

We propose to call such matrices "Manin matrices", since Yurii Manin first considered them at 1988-89. For such matrices basically all commutative facts holds true, despite they are quite far from commuative ones.

http://arxiv.org/abs/0901.0235

Another probably most famous examples are - "quantum group" matrices.
Here one requires ac=q cb, etc... for some number "q" ,
then there are q-determinant and also inverse can be written by the usual formula
substituting determinants by q-determinants...

There are many other examples super-matrices, Capelli matrices, matrices satisfying "reflection equation"... but the systematic theory is not developped.