When learning the condition number of matrices, defined as $K(A) = ||A||\cdot||A^{-1}||$, I found that there may be a contradiction about using it to judge whether a linear system question is an ill question, i.e. when $K(A)$ is a really big number, which means there may be a big error between $A^{-1}$ computed on computer and its real value, why can we believe that $K(A)$ can characterize the error of the solution of $Ax=b$?
Furthermore, I wonder whether we can estimate $K(A)$ by only using $A$ rather than $A$ and $A^{-1}$.