The inverse of $(A+A^{-1})$ when $A=A^{-1}$

Question

I have a matrix that is its own inverse, $A=A^{-1}$. I want to calculate the inverse of $(A+A^{-1})$, for which I would like to use the following chain of equalities: $$(A+A^{-1})^{-1}=(2A)^{-1}=2A^{-1}=2A$$

It doesn't seem to work though, why is that? I can't find anything related among the arithmetic rules in our course literature.

Answer 1

$A+A^{-1} = 2A$ The inverse of $2A$ is $2^{-1} A^{-1}$ i.e the inverse of $A+A^{-1}$ is $2^{-1} A^{-1}$

Answer 2

We just need to find a matrix $B$ so as $B(A+A^{-1})=I$.After some trials we see that $1/2A$ is what we are looking for.