Suppose that $n$ is even. Is it true that
$$\mathrm{SL}_n (\mathbb{R})/\{ \pm I \} \times \{\pm 1\} \cong \mathrm{GL}_n(\mathbb{R})/{\sim}$$
where $A\sim B$ if and only if $A=aB$ for some $a\in \mathbb{R}\setminus\{0\}$? I am asking about an isomorphism as multiplicative groups. Can someone give me a hint?