Inverse of the sum of matrix and its tranpose

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Is there some sort of identity or relation for getting the inverse of sum of a matrix and its transpose? Suppose I have a matrix A which is not symmetric and I want to get:

$$ B = (A + A^T)^{-1} $$

It would have been easier if $A$ is symmetric but I want to generalize some calculations to include directed networks.

Thank you very much.