with $X$ a symmetrix matrix of order $n$, $y$ a vector of length $n$ and $z$ a scalar, all to be determined so that:

$$\left[\begin{array}{cc}A& b\\ {b}^{T}& c\end{array}\right]\left[\begin{array}{cc}X& y\\ {y}^{T}& z\end{array}\right]=\left[\begin{array}{cc}I& 0\\ {0}^{T}& 1\end{array}\right]$$

where $I$ is the identity matrix of order $n$ and $0$ a vector of length $n$ with all components $0$.

The problem is still stated as:

$$\left[\begin{array}{cc}A& b\\ {b}^{T}& c\end{array}\right]\left[\begin{array}{cc}X& y\\ {y}^{T}& z\end{array}\right]=\left[\begin{array}{cc}I& 0\\ {0}^{T}& 1\end{array}\right]$$

However, in this case, $X$, $y$ and $z$ are known, and ${A}^{-1}$ has to be computed. To do that, compute:

Note that if
$z=0$
then
$A$
is not invertible and
${A}^{-1}$
does not exist.