# Determining order of matrices in $GL_2(\mathbb{F}_7)$

I need to determine the orders of the following matrices in the group $GL_2(\mathbb{F}_7)$: $$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 2 & 0\\ 0 & 1 \end{pmatrix}$$

Could someone provide me with a complete solution for the first matrix so that I know how to redo it with the second matrix?

Hint: n'th power of the first matrix will have n in (1,1) position. Take 7th power and use that elements belongs to $\mathbb{F}_7$. – tessellation Sep 16 '13 at 18:16
@tessellation So in $\mathbb{F}_7$, the above matrix equals the identity matrix – Jean-Francois Rossignol Sep 16 '13 at 18:20
But I’m glad you didn’t ask about $\pmatrix{1&3\\1&1}$. – Lubin Sep 16 '13 at 18:29
A partial answer good enough for your specific question: $$\pmatrix{1&m\\0&1}\pmatrix{1&n\\0&1}=\pmatrix{1&m+n\\0&1}\,,$$ and $$\pmatrix{a&0\\0&1}\pmatrix{b&0\\0&1}=\pmatrix{ab&0\\0&1}\,.$$ These relations are good no matter where (i.e. in what ring) the entries lie. So the question boils down to asking the additive order of $1$ and the multiplicative order of $2$ in $\mathbb F_7^+$ and $\mathbb F_7^\times$, respectively.