## How recursively calculate sum of elements of inverse of a PSD matrix

Suppose $A_{N \times N}$ and $B_{N+1 \times N+1}$ are two symetric positive definte matrix such that: $B=
\begin{bmatrix}
& & & & b \\
& A & & & b \\
& & & &b \\
b& b& ...&b& b^{'}
\end{bmatrix}$

Let use define operator $SUM$ for a matrix $G_{k \times k}$such that:
$SUM(G)=\Sigma_{i=1}^{k}\Sigma_{j=1}^{k}G_{ij}$ then:
$SUM(B^{-1})=\frac{SUM(A^{-1})+b^{'}-2bb^{'}SUM(A^{-1})}{1-b^{2}b^{'}SUM(A^{-1})}$
wher $G^{-1}$ means inverse of the matrix $G$