Computing inverse matrix when an element changes
The Sherman-Morrison formula could help:

$$ (A + uv^T)^{-1} = A^{-1} - \frac{A^{-1} uv^T A^{-1}}{1 + v^T A^{-1} u}. $$

Let $u = (a'_{ij}-a_{ij}) e_i$ and $v = e_j$, where $e_i$ is the standard basis column vector. You can check that if the updated matrix is $A'$ then
$$ A^{\prime -1} = A^{-1} - \frac{(a'_{ij}-a_{ij})A^{-1}_{i\rightarrow} A^{-1T}_{\downarrow j}}{1 + (a'_{ij}-a_{ij})A^{-1}_{ij}}.$$