If $A$ and$ I+AB$ are invertible, show $I+BA$ is also invertible

Question

Show that if $A$ and $I+AB$ are invertible, then $I+BA$ is also invertible with $$(I+BA)^{-1} = A^{-1}(I+AB)^{-1}A$$

Answer 1

Hint: Just do it. Use the fact that

$$ (I + BA) A^{-1} = A^{-1} + B = A^{-1}(I+AB).$$

to show that

$$ (I + BA) A^{-1} (I+AB)^{-1} A = I.$$

Answer 2

Since $\,A^{-1}C^{-1}A=(A^{-1}CA)^{-1}\,$ , we get with $\,C=I+AB\,$ :

$$(I+BA)\left(A^{-1}(I+AB)^{-1}A\right)=(I+BA)(A^{-1}(I+AB)A)^{-1}=$$

$$(I+BA)(I+BA)^{-1}=I$$

Answer 3

$$ (I + BA) = A^{-1}(I + AB)A \Rightarrow (I + BA)^{-1} = A^{-1}(I + AB)^{-1}A $$