We need the condition number of a matrix $\kappa(A)= \frac{\text{max}_{\vert\vert \vec{y} \vert\vert =1}\vert\vert A \vec{y} \vert\vert}{\text{min}_{\vert\vert \vec{y} \vert\vert =1}\vert\vert A \vec{y} \vert\vert}$ to help us understand how much the solution vector will change if there is a small change in the input matrix or output vector. It can also be used to see that there are rounding errors made in solving $A \vec{x} = \vec{b}$ when working with floating point numbers.
However, why would we need to know about the condition number of the inverse of $A$? Why is $\kappa({A^{-1}})$ important?
The question rather silly, given that by definition $\kappa(A)=\|A\|_2 \cdot \|A^{-1}\|_2 = \kappa(A^{-1})$.