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][1], 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)?  

 

  [1]: https://www.sciencedirect.com/science/article/pii/0024379586900157