TITLE: If $A$ and$ I+AB$ are invertible, show $I+BA$ is also invertible
QUESTION [7 upvotes]: Show that if $A$ and $I+AB$ are invertible, then $I+BA$ is also invertible with
$$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$

REPLY [3 votes]: $$
(I + BA) = A^{-1}(I + AB)A \Rightarrow (I + BA)^{-1} = A^{-1}(I + AB)^{-1}A
$$

REPLY [2 votes]: Since $\,A^{-1}C^{-1}A=(A^{-1}CA)^{-1}\,$ , we get with $\,C=I+AB\,$ :
$$(I+BA)\left(A^{-1}(I+AB)^{-1}A\right)=(I+BA)(A^{-1}(I+AB)A)^{-1}=$$
$$(I+BA)(I+BA)^{-1}=I$$