Suppose that $A$ is a positive definite matrix. Is it true that \begin{align} (I+A) A (I+A)^{-1}=A \end{align} I tried to use Woodbury identity and got to \begin{align} (I+A)( A -A (A^{-1}+I)^{-1}) \end{align}
2026-03-25 04:39:38.1774413578
Is $(I+A) A (I+A)^{-1}=A$ if $A$ is psd
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Hint:
$$(I+A)A=A(I+A)$$
from there, the identity should be obvious.