Show that if matrices A and B are elements of G, then AB is also an element of G. 
Let $G$ be the set of $2 \times 2$ matrices of the form
  \begin{pmatrix} a & b \\ 0 & c\end{pmatrix}
  such that $ac$ is not zero. Show that if matrices $A$ and $B$ are elements of $G$, then $AB$ is also an element of $G$.

Do I just need to show that $AB$ has a non-zero determinant? 

Proving that AB has a non-zero determinant is not enough, because not all 2x2 matrices with non-zero determinant are a element of G.
You need to prove another property of AB. This property is that it has the shape you stated.
This combined with a non-zero determinant guarantees that AB has the prescribed shape with ac not zero.
