What's the best algorithm to invert a matrix of non-commutative elements? In my case I have a matrix of matrices.

From first principals by equating the elements of M * M' to I (where M' is the inverse) I've worked out the inverse for a 2x2 Matrix (note that C, D is the 2nd row):

M = (A, B, C, D)

M' = ( (A-BD'C)', (C-DB'A)', -D'C(A-BD'C)', -B'A(C-DB'A)' )

(X' is the inverse)

Is is a matter of taking an existing algorithm - say LU Decomposition - and ensuring it respects non-commutativity or is there some more subtle maths involved?

Thanks

Alan