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

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