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Let $T$ be an invertible linear operator on $\mathbb{R}^2$. Is it true that if $T$ has determinant $\pm 1$ , then $T$ and $T^{-1}$ have the same norm (the usual norm operator)?

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    Note that this is not true for matrices larger than $2 \times 2$, e.g. the diagonal matrix with diagonal elements $[2,2,1/4]$ and its inverse don't have the same norm.2012-07-20

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