I'm not sure if you'll see this 2 days later, but I've been trying to make heads or tails of your example and I think there’s a mistake in there somewhere.

Maybe I'm hopelessly lost, but, for one, the inverse of

    ( 1  1  1)
    ( 1  2  3)
    ( 1  2  4)

is not:

    ( 2 -1  0)
    (-3  2 -1)
    ( 1 -2  1)

On the off chance you see this, any pointers?