For simple 2 x 2 matrices, the inverse of
a b
c d
is
d -b
-c a
divided by the determinant: ad-bc.
Why don't you treat your 2x2 partitioned matrix as though it were a simple 2x2? You will be surprised how easily the inverse of A appears.
I don't really know how to go about solving this problem, since it's a partitioned matrix. If I write it out in its complete 4 * 4 form, it will take a long time to reduce it, and I won't be able to get an answer in the form required by the question. So what method of inverse should I use?