Let $H=\left\{\left( \begin{matrix} a&b \\0&d\end{matrix} \right) :a,b,d\in \Bbb R \text{ and } ad\neq 0\right\}$. Show that $(H,\cdot)$ is a normal subgroup of $GL_2(\Bbb R)$.
Is this a correct statement? I mean $\left( \begin{matrix} 2&1 \\1&1\end{matrix} \right) \in GL_2(\Bbb R)$ and in order to have
$$\left(\begin{matrix} 2&1 \\1&1\end{matrix} \right)\left(\begin{matrix} a&b \\0&d\end{matrix} \right)\left(\begin{matrix}1&-1 \\-1&2\end{matrix} \right)\in H$$ shouldn't it satisfy $a-b-d=0$ so $a=b+d$?