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Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. Denote by $C$ the square block matrix given by $$ C=\left[\begin{array}{cc} 0 & B\\ A & 0 \end{array}\right]. $$ I am looking for the inverse $$ D=\left(I-C\right)^{-1} $$ where $I$ is the identity matrix. There are some nice equations for the inverse of $2\times 2$ block matrices and for block diagonal matrices, but I'm not sure how to tackle matrices like $I-C$. (see also how to invert anti-diagonal block matrices)

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The matrix $I-C$ is a block matrix with 2 invertible matrices on the diagonal. You can use the equation that you are linking to by substituting an $I$ of the appropriate size for $A$ and $D$, $-A$ for $B$ and $-B$ for $C.$

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  • $\begingroup$ Duh, not sure why I didn't see that. Thank you! $\endgroup$ Commented May 15 at 20:56
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    $\begingroup$ Sorry, had to apply a correction because I mixed up the letters $\endgroup$
    – Lieven
    Commented May 15 at 20:57

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