# Inverse of $(rP+\bar{r}\bar{P})$ where $P=P^*$ and $r=\exp(i\theta)$

I have a positive definite Hermitian matrix $$P=P^*>0$$ where $$P^*$$ is the conjugate transpose of $$P$$ and $$r=\exp(i\theta)$$. So, how can I prove that $$(rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$ Where, $$Y$$ is a positive definite Hermitian matrix and $$\bar{Y}$$ is its conjugate?