Show that $T$ is normal in $GL_{2}(\Bbb{R})$.

Question

Show that $T=\left \{ \begin{bmatrix} a &b \\ 0 & c \end{bmatrix}:a,b,c\in \Bbb{R}, ac\neq 0 \right \}$ is normal in $GL_{2}(\Bbb{R})$.

I don't think it is normal, because for any arbitrary matrix $a\in GL_{2}(\Bbb{R})$ and $t\in T$, $ata^{-1}\notin T$. Is this a valid argument?