The impact of minor change in matrix inverse on $A A^{-1}$

Question

What happens if we change one element of the inverse matrix $A^{-1}$ and then multiply this new matrix with the original matrix $A$? Any proofs?

Answer 1

Let $\rm B := A^{-1}$. Suppose the $(i,j)$-th entry of $\rm B$, which we denote by $b_{ij}$, is changed to $c_{ij}$. Hence,

$$\mathrm B^{-1} \left( \mathrm B + (c_{ij} - b_{ij}) \, \mathrm e_i \mathrm e_j^\top \right) = \mathrm I + (c_{ij} - b_{ij}) \, \mathrm B^{-1} \mathrm e_i \mathrm e_j^\top = \mathrm I + (c_{ij} - b_{ij}) \, \mathrm a_i \mathrm e_j^\top$$

where $\mathrm a_i$ denotes the $i$-th column of $\rm A$.