Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.

$(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)^{-1},(A_1+A_2+A_3)^{-1}$

My main limitation is memory, i.e. I don't want to store the $2^c$ inverses, but find a way given a limited number of inverses to be able to compute the inverse I want from those I stored.

To go a bit farther, the matrices will be continuously updated using rank-one updates. Thus I would like to compute the rank-one updates of a limited subset of these $2^c$ inverses and be able to compute the inverse I want in $O(n^2)$.

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Could you clarify a bit? Do you really need to compute all the $2^c$ inverses? Is that exponential time acceptable, because later you say that "...to compute the inverse from those stored.."---because it seems better to really just compute the desired inverse directly. – Suvrit Sep 1 '13 at 2:42
I will access them all multiple times and dont mind paying $O(n^2)$ each time I need an inverse. My main problem is I dont have the RAM to store all $2^c$ inverses. – Leo Sep 1 '13 at 11:57