Normal Subgroups and Factor Groups

Let G = GL(2,R) and let K be a subgroup of R*. Prove that H = {A ∈ G| det A ∈ K} is a normal subgroup of G.

A subgroup is said to be normal iff we have

So take it is a group and therefore invertible so but consider for any we would have:

so we see that for any and for all we get .

This shows thus it is normal.