I am trying to use the Woodbury formula  in order to apply corrections on the inverse of a matrix A I already calculated once as X (rather big: n=6000+). Once in a while I would like to compute the inverse of A with a "k" changes ranging from 2-3 to 20.
Applying the woodbury formula should be of big help there, and it works perfectly when I only apply 2 or 3 changes. But when I apply 20, one of the matrix I need to invert in the calculation (kind of known as the capacitance matrix ) becomes singular (determinant gets way too small). I am thus unable to apply the formula. Note that the corrections applied are valid, i.e. if I create the new matrix from scratch, it is non-singular.
My questions are the following: - is it something I should expect? - are there anyway to solve it?
I have been looking everywhere for practical information regarding woodbury's usage in a numerical analysis context but can not find much information. I would be very grateful for any guidance on that topic.