Compute the inverse of a matrix with partially known values I have an odd problem. I have a matrix $A$ of size 16 per 16. As an example, we will consider the 4 by 4 matrix below.
$$A = \begin{pmatrix}50 & 3 & 10 & 2\\\ 3 & 60 & 7 & 1\\\ 10 & 7 & 55 & 4\\\ 2 & 1 & 4 & 45 \end{pmatrix}$$
I want to find the matrix $B$ such as:
$$B = \begin{pmatrix}b_{11} & b_{12} & b_{13} & 0\\\ b_{21} & b_{22} & 0 & 0\\\ b_{31} & 0 & b_{33} & b_{34}\\\ 0 & 0 & b_{43} & b_{44} \end{pmatrix}$$
where I forced some of the values $b_{ij}$ to $0$ and where $C = B^{-1}$ such that $C_{ij}$ is equal to (or as close as possible) to $A_{ij}$ if $B_{ij} \neq 0$.
In other words, if I inverse the matrix $B$, I want to get the same values as $A$ where in the position $(i, j)$ $B$ is not set to $0$.
Additionally, the matrix $A$ and the matrix $B$ are symmetric. Moreover, the diagonal values $A_{ii}$ are about 10 times larger than the off-diagonal values $A_{ij}$. 
Is there a way to solve this problem?
N.B: If anyone has a better title for this post... I'm open to suggestions :) 

We have $7$ equations in $7$ variables $b_{11},b_{12},b_{13},b_{22},b_{23},b_{33},b_{34}$, which have, for example, a solution
$$
B=\begin{pmatrix} 
\frac{27497}{1321025} & - \frac{1}{997} & - \frac{1}{265} & 0 \\\
- \frac{1}{997} & \frac{50}{2991} & 0 & 0 \\\
- \frac{1}{265} & 0 & \frac{136093}{7167985} & - \frac{4}{2459} \\\ 
0 & 0 &  - \frac{4}{2459} & \frac{55}{2459}
\end{pmatrix},
$$
where $B^{-1}$ has the required form. In fact, we have
$$
B=\begin{pmatrix} 
50 & 3 & 10 & \frac{8}{11} \\\
3 & 60 & \frac{3}{5} & \frac{12}{275} \\\
10 & \frac{3}{5} & 55 & 4 \\\
\frac{8}{11} & \frac{12}{275}  & 4 & 45
\end{pmatrix}
$$
