My question is motivated by the inverse of $A^{-1}+B^{-1}$.
$$A^{-1}+B^{-1}=A^{-1}(A+B)B^{-1}\implies(A^{-1}+B^{-1})^{-1}=B(A+B)^{-1}A$$
$$A^{-1}+B^{-1}=B^{-1}+A^{-1}=B^{-1}(A+B)A^{-1}\implies(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$$
$$B(A+B)^{-1}A=A(A+B)^{-1}B$$
Is this a special result due to the fact that $(A+B)^{-1}$ is sandwiched between $A$ and $B$, or does it hold for other cases as well, i.e. $AXB=BXA$ where $X$ has some special properties?
If it's the former, then some intuition for why it holds beyond the math mechanics that I have shown above would be appreciated.