I need to show that the quotient group $G/H$ is isomorphic to $\mathbb{R}×$; where $G = \{ A = (a b 0 d): \det A \neq 0 \}$, $H = \{B = (a b 0 a): \det B \neq 0\}$, $\mathbb{R}\times$ is the group of non-zero real numbers under multiplication.
Attempt: I've already proven that $H$ is normal in $G$. Is $G$ isomorphic to $\mathbb{R}\times$? I tried the map $f: A \to \det A$ but $f$ is not 1-1.
Hint Consider $f$ defined on $G\rightarrow R-\{0\}$ defined by $f(\pmatrix{a &b\cr 0&d})=a/d$, $f(\pmatrix{ a& b\cr 0&d})=ad=1$ is equivalent to saying that $a=d$ i.e $ker f=H$.