<s>[INST] Find the order of the matrices $A$ and Matrice $B$? 
Find  the   order of the  matrices $A$ and $B$ in the Group $GL_2(\mathbb{F}_7)$: $$A=\begin{bmatrix}  1 &1 \\ 0& 1 \end{bmatrix}\;\text{and}\; B=\begin{bmatrix}  2 &0 \\ 0 & 1 \end{bmatrix}$$ 

For $A$ I take  $
\pmatrix{1&x\\0&1}\pmatrix{1&y\\0&1}=\pmatrix{1&x+y\\0&1}\,,
$
and
for $ B$ we have 
$
\pmatrix{a&0\\0&1}\pmatrix{b&0\\0&1}=\pmatrix{ab&0\\0&1}\,.
$
After that  im not able to proceed  further ,
any hints/solution 
thanks in advance
 [/INST] For your matrix $A$, we have $$\begin{pmatrix}1 &1 \\0 &1 \end{pmatrix}^n=\begin{pmatrix}1 &n \\0 &1 \end{pmatrix}$$
so, we have to find a least $n$ for which $$\begin{pmatrix}1 &n \\0 &1 \end{pmatrix}=\begin{pmatrix}1 &0 \\0 &1 \end{pmatrix}$$ Now, clearly for $n=1,2,\cdots,6$ it does't happen but in the seventh power it becomes identity. 
So $$\text{order of}\;\begin{pmatrix}1 &1 \\0 &1 \end{pmatrix}=7$$
In general order of $\begin{pmatrix}1 &1 \\0 &1 \end{pmatrix}$ in $GL(2,\Bbb{Z}_p)$ is $p$ where $p$ is prime
Similarly work for the second matrix
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