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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 ${A_{ii}^{-1}}$?

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  • $\begingroup$ how do you know that $A$ is invertible? $\endgroup$ – abel Apr 6 '15 at 17:56
  • $\begingroup$ It is assumed that $A^{-1}$ exists. $\endgroup$ – John Smith Apr 6 '15 at 17:57
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    $\begingroup$ Do you mean $A_{ij}$ are matrix blocks, each a constant matrix, with a different constant for each block, and with many zero blocks so that the matrix is sparse? Then my previous comment does not hold. $\endgroup$ – Jean-Claude Arbaut Apr 6 '15 at 18:01
  • $\begingroup$ @Jean-ClaudeArbaut, each off-diagonal block is a constant matrix. Some are zero matrices, and some are not. $\endgroup$ – John Smith Apr 6 '15 at 18:03
  • $\begingroup$ Ok, thanks, I didn't understand the question at first. $\endgroup$ – Jean-Claude Arbaut Apr 6 '15 at 18:05

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