Suppose $\displaystyle A_{N \times N}$ and $\displaystyle B_{N+1 \times N+1}$ are two symetric positive definte matrix such that: $\displaystyle B=
\begin{bmatrix}
& & & & b \\
& A & & & b \\
& & & &b \\
b& b& ...&b& b^{'}
\end{bmatrix}$
Let use define operator $\displaystyle SUM$ for a matrix $\displaystyle G_{k \times k}$such that:
$\displaystyle SUM(G)=\Sigma_{i=1}^{k}\Sigma_{j=1}^{k}G_{ij}$ then:
$\displaystyle SUM(B^{-1})=\frac{SUM(A^{-1})+b^{'}-2bb^{'}SUM(A^{-1})}{1-b^{2}b^{'}SUM(A^{-1})}$
wher$\displaystyle G^{-1}$ means inverse of the matrix $\displaystyle G$