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

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

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.

• To get a feeling for what you mean by "explicit inverse", how would a possiblbe answer for $1\times 1$ matrices look? – Henning Makholm Aug 17 at 2:40
• 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? – hardmath Aug 17 at 2:41
• For example, if $A=I$ (diagonal) and $B=-I$ (symmetric), then $I+AB=0$ is not invertible. – alex.jordan Aug 17 at 4:12
• One has to provide that $-1$ is not an eigenvalue of $AB$. – Michael Hoppe Aug 17 at 4:46
• Thank you all for kind feedback. More conditions are added to guarantee the existence of the inverse :-) – Jack Xu Aug 18 at 3:28