Question #165252

Describe (list the elements, give the identity and inverses) the cyclic


group generated (under multiplication) by [ 1 1 ]

[0 1] matrix



1
Expert's answer
2021-02-24T07:48:46-0500

Solution:

Let given matrix be A=[1101]A=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}

Now, A2=[1101]2=[1201]A^2=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}^2=\begin{bmatrix}1&2\\ 0&1\end{bmatrix}

Next, A3=A2.A=[1201].[1101]=[1301]A^3=A^2.A=\begin{bmatrix}1&2\\ 0&1\end{bmatrix}.\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}1&3\\ 0&1\end{bmatrix}

And so on.

Then, we observe that An=An1.A=[1n101].[1101]=[1n01][1001]A^n=A^{n-1}.A=\begin{bmatrix}1&n-1\\ 0&1\end{bmatrix}.\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}1&n\\ 0&1\end{bmatrix}\ne\begin{bmatrix}1&0\\ 0&1\end{bmatrix}

We see that for no value of nn , we have An=IA^n=I .

Thus, given matrix A is not cyclic.



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