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Possible Duplicate:
How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I know $\rm{GL_n}(\mathbb{R})$ is not connected and connected components are $C_1 =\{A:\rm{det}\ A>0\}$ and $C_2=\{A:\rm{det} \ A<0\}$.

Given that $C_1= \rm{det}^{-1}(0,\infty)$ and $C_2=\rm{det}^{-1}(-\infty,0)$, $\rm{det}:M_n(\mathbb{R})\to \mathbb{R}$.

But how can one prove that $C_1$ and $C_2$ are connected in $\rm{M_n}(\mathbb{R})$?

Thank you.

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    Think of $M_n(\mathbb{R})$ as $\mathbb{R}^{n\times n}$.2012-07-17
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    @Davide: Sorry, I hadn't refreshed to see your duplicate before I posted the answer. I've now voted to close as duplicate.2012-07-17
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    could any one explain me in a little detail about the point (ii) of the fisrt answer of the question?2012-07-18

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