When the inverse matrix is known at order n, compute it at order n+1

That is to say: add one row-and-column and update the inverse.

Note that if $z=0$ then $A$ is not invertible and ${A}^{-1}$ does not exist.
Remark: here one row-and-column is attached or eliminated in the last position only for convenience of presentation.
The inverse matrix can be updated in this way when any row-and-column is eliminated.
When a row-and-column is attached in the last position, row and columns can be sorted afterwards (in both matrices).
Francesco Uboldi 2014,2015,2016,2017