Prove $\operatorname{GL}_n(\Bbb{R})/\operatorname{SL}_n(\Bbb{R}) \cong \Bbb{R}^\times$ $$\operatorname{GL}_n(\Bbb{R})/\operatorname{SL}_n(\Bbb{R}) \cong \Bbb{R}^\times$$
This is trivial to prove with the first isomorphism theorem - by using the determinant as a endomorphism, then as $\operatorname{SL}_n(\Bbb{R})$ is $1$ under the determinant, it is the kernel, and by FIT, the above is proved.
I was wondering how to prove this without the isomorphism theorem ?

Hint: consider the homomorphism 
$$\mathbf{R}^\times \ni \lambda \mapsto (\lambda I_n) SL_n \in GL_n/SL_n$$
where $I_n$ is the $n\times n$ identity matrix and $gSL_n$ is the coset of $g$ in the quotient group.
