find an element of order p in GL2(Zp)

Question

I am kind of stuck on one of my homework questions, which asks:

Let $\mathbb{Z}_p$ denote the integers modulo $p$. Find an element of order $p$ in $\mathbb{GL}_2(\mathbb{Z}_p)$. Can you also find an element of order $2p?$

I understand the concept of general linear group and modulo class but I am in need of some hints about finding these kinds of element.

Thanks a lot.

Answer 1

Hint:

1. Consider matrices of the form $\pmatrix{ 1 & b \\ 0 & 1}$.

2. Consider matrices of the form $\pmatrix{ -1 & \hphantom{-}b \\ 0 & -1}$.

Handle $p=2$ separately.