# a special matrix inverse

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

• how do you know that $A$ is invertible? – abel Apr 6 '15 at 17:56
• It is assumed that $A^{-1}$ exists. – John Smith Apr 6 '15 at 17:57
• 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. – Jean-Claude Arbaut Apr 6 '15 at 18:01
• @Jean-ClaudeArbaut, each off-diagonal block is a constant matrix. Some are zero matrices, and some are not. – John Smith Apr 6 '15 at 18:03
• Ok, thanks, I didn't understand the question at first. – Jean-Claude Arbaut Apr 6 '15 at 18:05