Let

$$M = \begin{bmatrix}
A & B \\
B & C
\end{bmatrix}$$
in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single term:
$$A =
\begin{bmatrix}
\sum_x \sum_y A(x,y) x^2 & \sum_x \sum_y A(x,y) xy & \sum_x \sum_y A(x,y) x\\
\sum_x \sum_y A(x,y) xy & \sum_x \sum_y A(x,y) y^2 & \sum_x \sum_y A(x,y) y\\
\sum_x \sum_y A(x,y) x & \sum_x \sum_y A(x,y) y & \sum_x \sum_y A(x,y) \\
\end{bmatrix}$$
$B$ and $C$ are exactly the same as $A$ but replacing $A(x,y)$ by $B(x,y)$ and $C(x,y)$ respectively.
If we assume every matrix is non-singular, is there an analytic way to get the diagonal of the inverse of matrix M, that is $M^{-1}_{i,i}$?

Thanks in advance!