You need to show if R is antisymettric, meaning if a<=b and b<=a implies a=b. Then $\displaystyle R^{-1}$ has the same property. It should be clear how to do this by definition.
If $\displaystyle (x,y) \in R^{ - 1} \wedge (y,x) \in R^{ - 1} \quad \Rightarrow \quad (y,x) \in R \wedge (x,y) \in R.$
But we know that $\displaystyle R$ is antisymmetric so $\displaystyle x=y$.