It is given that $G=\left\{\begin{pmatrix} a & b\\ 0 & c \end{pmatrix}\text{ with }a,c \in \mathbb Q^∗\text{ and }b \in \mathbb Q\right\}$ is a subgroup of $GL_2(\mathbb Q)$, the group of invertible 2×2-matrices with rational coefficients.
$N=\left \{\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix}\text{ with }a \in \mathbb Q^∗\text{ and }b \in \mathbb Q\right\}$.
Show that N is a normal subgroup of G and that $G/N$ is isomorphic to $\mathbb Q^∗$, which is a group under multiplication.
My attempt: N is a normal subgroup of G if $gNg^{-1} = N$
$\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix}$$\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix}$$\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix}^{-1}$ = $\begin{pmatrix} a & ab+b\\ 0 & 1 \end{pmatrix}$$\begin{pmatrix} 1/a & -b/a\\ 0 & 1 \end{pmatrix}$ =$\begin{pmatrix} 1 & ab\\ 0 & 1. \end{pmatrix}$ = N if $a = 1$.
Then to prove that $G/N$ is isomorphic to $\mathbb Q^∗$ , define the map $\varphi$: $G/N \rightarrow \mathbb Q^∗ $ by $\varphi\left(\begin{pmatrix} a & b\\ 0 & c. \end{pmatrix}\right)= a$.
I know that $G/N$ is isomorphic to $\mathbb Q^∗$ iff the map is surjective and homomorphic. It is homomorphic since $$\begin{align*}\varphi\left(\begin{pmatrix} a & b\\ 0 & c. \end{pmatrix}\begin{pmatrix} d & e\\ 0 & f. \end{pmatrix}\right) &= \varphi\left(\begin{pmatrix} ad & ae+bf\\ 0 & cf. \end{pmatrix}\right)\\ &=ad\\ &=\varphi\left(\begin{pmatrix} a & b\\ 0 & c. \end{pmatrix}\right) \varphi\left(\begin{pmatrix} d & e\\ 0 & f. \end{pmatrix}\right).\end{align*}$$
But I don't know how to prove that it is surjective. Could please someone explains to me how to do it?