I am not able to prove or find a counter example for the following statement.

Let $b:\mathbb{R}^n\rightarrow GL(n,\mathbb{R})$ be such that for $C>0$

$\frac{1}{C}\vert \xi\vert^2\leq \xi^Tb(x)\xi\leq C\vert \xi\vert^2$

for all $\xi,x\in\mathbb{R}^n$, i.e., $b$ is uniformly elliptic and bounded above. Does the map $x\mapsto b(x)^{-1}$ satisfy a similar property?

I have been thinking about this for quite a while now but I cannot think of a proof or a counter example. I would be very grateful if someone shares his ideas with me.

Thank you.


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