Answer to Question #165252 in Abstract Algebra for K

Question

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

group generated (under multiplication) by [ 1 1 ]

[0 1] matrix

Answer 1

Solution:

Let given matrix be $A=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}$

Now, $A^2=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}^2=\begin{bmatrix}1&2\\ 0&1\end{bmatrix}$

Next, $A^3=A^2.A=\begin{bmatrix}1&2\\ 0&1\end{bmatrix}.\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}1&3\\ 0&1\end{bmatrix}$

And so on.

Then, we observe that $A^n=A^{n-1}.A=\begin{bmatrix}1&n-1\\ 0&1\end{bmatrix}.\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}1&n\\ 0&1\end{bmatrix}\ne\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$

We see that for no value of $n$ , we have $A^n=I$ .

Thus, given matrix A is not cyclic.