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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.