(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The following matrix is an elements of the group GL_{2}(2), that is, the general linear group of 2x2 matrices in [tex]\mathbb{Z}_2[/tex]:

[tex]A = \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix}[/tex]

Find the order of the element A.

3. The attempt at a solution

I know that the order of A is 3. Because A^{3}=I, where "I" is the identity. I found this by trial and error:

[tex]A^1 = \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix}^1 \neq \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}[/tex]

[tex]A^2 = \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix} \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix} \neq \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}[/tex]

[tex]A^3 = \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix} \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix} \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix} = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}[/tex].

Here is my question, is there a shorthand method for finding "n" in:

[tex]\begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix}^n = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}[/tex]

Is there any way of solving for n without going through all the suffering matrix multipications above?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Matrix Order

**Physics Forums | Science Articles, Homework Help, Discussion**