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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?

Thank you in advance

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All you need to do is find the first power of each matrix which equals identity. In this case, one can find an easy formula for powers of both matrices (find the first couple of powers and it should become clear). –  Jonathan Y. Sep 16 '13 at 18:15
    
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
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@tessellation So in $\mathbb{F}_7$, the above matrix equals the identity matrix –  Jean-Francois Rossignol Sep 16 '13 at 18:20
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@user43208 Then the order is 7. For the second one, the order is 3 –  Jean-Francois Rossignol Sep 16 '13 at 18:26
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But I’m glad you didn’t ask about $\pmatrix{1&3\\1&1}$. –  Lubin Sep 16 '13 at 18:29
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1 Answer

up vote 4 down vote accepted

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.

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