Proof of Matrix Norm (Inverse Matrix)
Solution 1:
Use $\lVert AB\rVert \leq \lVert A\rVert \lVert B\rVert$, as the induced norm is in particular submultiplicative. So that $\lVert I_n\rVert \leq \lVert A\rVert \lVert A^{-1}\rVert$.
Solution 2:
Suppose $|y|=1$ is such that $|A|=|Ay|$. Then $x=Ay/|A|$ also has norm $1$ so it follows that $$ |A^{-1}|\geq |A^{-1}x|=|A^{-1}Ay/|A||=\frac{|y|}{|A|}=\frac{1}{|A|}=|A|^{-1}. $$