Condition numbers of invertible 2x2 matrices

Question

After learning about condition numbers, I worked through some MATLAB examples to compute condition numbers of several 2x2 matrices to gain some intuition. I noticed that for invertible 2x2 matrices, the 2-norm condition number seems to always be equal to the infinity-norm condition number. My question is twofold, does this result hold in general, and how can this be proven?

Answer 1

If $A \in \mathbb{R}^{m \times n}$ then you have the following bound

$$ \frac{1}{\sqrt{n}}\| A\|_{\infty} \leq \| A\|_{2} \leq \| A \|_{\infty}$$

In this case $n=2$ so

$$ \frac{1}{\sqrt{2}}\| A\|_{\infty} \leq \| A\|_{2} \leq \| A \|_{\infty}$$