Subgroups of $GL(2, \Bbb{R})$

Question

I am wondering if a subgroup of $GL(2,\Bbb{R})$ which is constructed by all rotations and all matrices in the form of $$ \left[ \begin{array}{l l} a & x \\ 0 & \sqrt{a} \end{array} \right] \ \ \ (a \in \Bbb{R}^+, x \in \Bbb{R}) $$ would be a proper subgroup of $GL(2,\Bbb{R})^+:=\{a \in GL(2, \Bbb{R}):\ \det(a)>0\}$ or it gives us the whole group?

Answer 1

The determinants of all your matrices in your subgroup would be positive, so it can't be all of $GL(2,\mathbb{R})$. If you quotient by the subgroup I'm not sure whether you get $\{-1,1\}$, or whether there is more stuff missing from your subgroup.