Proving $||A^{-1} v || \geq ||A^{-1} u ||$ implies $||v|| \geq ||u||$.
Solution 1:
This is not possible. Let $A^{-1}v = \lambda v$ and $A^{-1}u = \mu u$. Then what you are trying to prove is $\lambda \lVert v \rVert \geq \mu \lVert u \rVert$ implies $\lVert v \rVert \geq \lVert u \rVert$, which is obviously not true for any norm.