Given an invertible matrix $A$ and column vectors $b$ and $c$.

For which $A$,$b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and $A^{-1}$ equal?

According to a result by Loewy, this is true if $B$ and $A^{-1}$ are diagonally similar with transpose (plus some extra conditions).

My main interests are:

1. For which matrices $A$ is the problem solvable?
2. Given a matrix $A$, how attain $b$ and $c$ numerically (best possible if no exact solution exist)?