If $A$ is a real symmetric singular matrix (similar to a Laplacian matrix, which comes from $M^{T}GM$, where M is incidence matrix and G is a diagonal matrix).
G has large values compared to $B$, which is a diagonal matrix.
then the inverse of $A-B$ is coming to be $k$ times $Q$, where $k$ is a scalar which is the reciprocal of the sum of elements of $B$, and $Q$ is a matrix with all elements approximately $1$. How to prove it mathematically?
i.e. $(M^{T}GM-B)^{-1}= k.\pmatrix{1+e1 &1+e2&1+e3\\1+e4 &1+e5&1+e6\\1+e7 &1+e8&1+e9}$,
where e1...e9 are very small.
$k= \frac{1}{\sum B}$