Groups under Multiplication

Groups under Multiplication

Let G=GL(2,R) and $ H = \left[ \begin{array}{ccc|c} a & 0 \\ 0 & b
\end{array} \right]$ |a and b are nonzero integers: under the operation
matrix multiplication. Disprove that H is a Subgroup pg GL(2, R).
Well I have learned that I have to prove that:
1)Show that e¸H (where e is the identity)
2)Assume that a¸H , b¸H
3)Show that a.b¸H
4)Shove that $(a.b)^-1$ (Inverse)
So I know that It does not hold, But how do I prove that?
When I try to prove that e¸H I only get the Identity matrix and that
holds because I get that a and b are nonzero integers.
When I prove that a and b is in the set I get that it is because both a
and b are nonzero integers and that is the identity matrix.
Now I proved that a.b¸H as follows:
$$ H = \left[ \begin{array}{ccc|c} a & 0 \\ 0 & b \end{array} \right]
\left[ \begin{array}{ccc|c} c & 0 \\ 0 & d \end{array} \right] = \left[
\begin{array}{ccc|c} ac & 0 \\ 0 & bd \end{array} \right] $$ With a = b =
c = d = 1 I get the Identity matrix again.
Now I know if a = b = 2 the subgroup will not hold because the inverse
will be a set of rational numbers and H is only a subgroup if it contains
only Integer.
Is my reasoning correct and if not where did I go wrong?