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Question. Is it possible to find the explicit inverse of $I+AB$, where $A$ is diagonal and $B$ symmetric matrix?

$A$, $B$ are non-singular, and $I+AB$ is invertible. For $B=S-I$, where $S=[s_{ij}]=[c*sinc(c\pi(i-j))]$, $i$ and $j$ are row and column indices, $sinc(x)=sin(x)/x$, $0<c<1$

It is equivalent to find the explicit inverse of $A_0+B$ where $A_0$ is diagonal and $B$ symmetric matrix.

Thanks for any feedback.

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    $\begingroup$ To get a feeling for what you mean by "explicit inverse", how would a possiblbe answer for $1\times 1$ matrices look? $\endgroup$ – Henning Makholm Aug 17 at 2:40
  • $\begingroup$ The given conditions do not imply that an inverse exists, so it seems the short answer is no. Is there more information about $A,B$, especially about their positivity? $\endgroup$ – hardmath Aug 17 at 2:41
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    $\begingroup$ For example, if $A=I$ (diagonal) and $B=-I$ (symmetric), then $I+AB=0$ is not invertible. $\endgroup$ – alex.jordan Aug 17 at 4:12
  • $\begingroup$ One has to provide that $-1$ is not an eigenvalue of $AB$. $\endgroup$ – Michael Hoppe Aug 17 at 4:46
  • $\begingroup$ Thank you all for kind feedback. More conditions are added to guarantee the existence of the inverse :-) $\endgroup$ – Jack Xu Aug 18 at 3:28

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