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Let $A$ be a real $2\times 2$ matrix such that $\det A=1$, show that $\|A\|=\left\|A^{-1}\right\|$.

Any hint would be appreciated, thanks.

EDIT: $\|\cdot\|$ is the operator norm $\|A\|=\max_{\|x\|=1}\|Ax\|$, all vector norms are Euclidean norms.

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    What is $\|\cdot \|$?2012-12-15
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    op norm, added an edit2012-12-15
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    Uses the facts: (1) $det(I_{n\times n})=1$, (2) $A^{-1}A=AA^{-1}=I_{n\times n}$, (3) $\|A B\|\leq \|A\|\cdot\| B\|\leq |trace(A^TB)|$ (4), (5) $det(I_{n\times n})=\|I_{n\times n}\|=1$ and (6) $trace(I_{n\times\n})=1$.2012-12-15
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    Don't you mean $tr(I_n)=n$?2012-12-15

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