Let
$A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an invertible matrix,
where
1) the elements in each off-diagonal block $A_{ij} \quad (i\neq j)$ have the same values, and
2) the elements in each diagonal block $A_{ii}$ are not the same values.
3) all elements in $A$ are non-negative,
4) $A$ is a sparse matrix.
Is there an easy way to find the inverse of the matrix $A$, given the inverse of each off-diagonal block ${A_{ii}^{-1}}$?