On the sharpness of an inequality involving operator norm of invertible matrices

Question

Let $A\in GL_n(\mathbb R)$.

How to show that for every $\delta >0$, $\exists 0\ne r,b \in \mathbb R^n$ such that $\delta=||r||_2/||b||_2$ and

$\dfrac {||r||_2}{||A^{-1}||_2||A||_2||b||_2}=\dfrac{||A^{-1}r||_2}{||A^{-1}b||_2}$ ?

NOTE: In general, I can prove that $\dfrac {||r||_2}{||A^{-1}||_2||A||_2||b||_2}\le \dfrac{||A^{-1}r||_2}{||A^{-1}b||_2}, \forall r,b\in \mathbb R^n \setminus \{0\}$ .

Answer 1

Just take $r$ as $\delta$ times the unit left singular vector for the largest singular value of $A$ and $b$ as the unit left singular vector for the smallest singular value of $A$.