$G = GL\_n(\mathbb{R})$ and $H$ is the subgroup of all matrices in $G$ with positive determinant

Is $H$ is a normal subgroup of $G$?

My attempt : Take G= $ \begin{bmatrix} 2 &0 \\ 0& 1 \end{bmatrix}$ and 
 $H=\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix} $

H will not normal subgroup of G 

Is its true ?