MHB This suggests that both H and K are normal subgroups of L. What am I missing?

[11hannab]

Question about matrix groups and conjugate subgroups?

This question concerns the group of matrices
L = { (a 0)
(c d) : a,c,d ∈ R, ad =/ 0}
under matrix multiplication, and its subgroups
H = { (p 0, (p - q) q) : p,q ∈ R, pq =/ 0} and K = { (1 0, r 1) : r ∈ R}Show that one of H and K is a normal subgroup of L and that the other is not.

Any pointers would be fantastic! Struggling and is from a previous assignment but just can't seem to figure this out?!
I understand that klk^-1 gives a lower triangular matric similar to L. Which would be this is a subgroup of L

But when i multiply together lhl-1 is seem to also get a lower traingular matrix?

Any pointer to where i am going wrong?

 

[Evgeny.Makarov]

It seems to me that for every $a$, $c$ and $d$ and for every $r$ there exists an $r'$ such that
\[
\begin{pmatrix}a&0\\c&d\end{pmatrix} \begin{pmatrix}1&0\\r&1\end{pmatrix} = \begin{pmatrix}1&0\\r'&1\end{pmatrix} \begin{pmatrix}a&0\\c&d\end{pmatrix}
\]