Suppose that $A$ is a square sparse matrix that is diagonal apart from one entry
$$ A = \begin{pmatrix} a_{11} & & \\ & \ddots & \\ & & a_{nn} \end{pmatrix} $$ with all $a_{ii}$ non-zero apart from one entry. Otherwise, the values are quite large $a_{ii} \approx 10\rightarrow 1000$ range.
Can I use a Neumann series to approximate the inverse matrix $A^{-1}$? Or, is there any way to approximate the inverse?
My thoughts are the following: First note that strictly speaking there is no such inverse because your matrix is singular. But let $a_{i,i}$ be the zero entry on the diagonal. Then set $a_{i,i} = \epsilon$. If you think about the inverse of the matrix you get, it'll be the matrix with the entries $b_{j,j} = \frac{1}{a_{j,j}}$ on the diagonal. Now as $\epsilon \rightarrow 0$ all entries stay constant except $b_{i,i} \rightarrow \infty$. So in general the larger you choose $b_{i, i}$, the better your approximation will be.