Question about the operator norm on $\mathbb R^2$

Question

So here is my question,

I have to decide whether the following statement is true

Let $T$ be an isomorphism on $\mathbb R^2$. Then $$\|T\|=\frac{1}{\|T^{-1}\|}$$

I am pretty sure that the statement is false. Let T be an invertible operator of $\mathbb{R}^2$. Then the following disproves the claim: \begin{align*} \|T\| &:= \sup_{x\neq 0} \frac{\|T(x)\|}{\|x\|} \\ &= \sup_{y \neq 0} \frac{\|y\|}{\|T^{-1}(y)\|} \\ &= \sup_{y \neq 0} \frac{1}{\frac{\|T^{-1}(y)\|}{\|y\|}} \\ &= \frac{1}{\inf_{y \neq 0}\frac{\|T^{-1}(y\|}{\|y\|}} \\ &\neq \frac{1}{\|T^{-1}\|} \end{align*}

Answer 1

For example, let $T:=\pmatrix{1&0\\0&2}$ then $\|T\|=2$ and $\|T^{-1}\|=1$.