Let $A_n$ be an $n\!\times\!n$ matrix whose inverse is known and let $A_{n+1}$ be a regular $(n\!+\!1)\!\times\!(n\!+\!1)$ matrix obtained by adding an $(n\!+\!1)$'st row and an $(n\!+\!1)$'st column to $A$, so it is "augmented" both on the right and at the bottom. Can knowing $A_n^{-1}$ help to make inversion of $A_{n+1}$ more efficient than it would be if we inverted $A_{n+1}$ straight-up, without knowing anything about $A_n$? This would be useful in a problem where matrices augmented in this way have to be inverted one after another.
I have come across something called Block Recursive Inversion, but have not studied it. Does that help with my question?
Will appreciate literature recommendation.
Thanks