Question

(b) Show that ab has infinite order.

Short answer

It is shown that, ab has infinite order.

Step-by-step solution

Determine given matrices

Consider the given matrices,

$a=\left(\begin{array}{cc}0& 1\\ -1& -1\end{array}\right)$has order 3 and$b=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ has order 4.

Determine ab has infinite order

Consider the ab, then:

$\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$

For any $r,s\in \mathbf{ℝ}$.

$\left[\begin{array}{cc}1& 0\\ r& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ s& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ r+s& 1\end{array}\right]$

Then it indicates that, $n\in \mathbf{\mathbb{N}}$,

Therefore,

${\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]}^n=\left[\begin{array}{cc}1& 0\\ -n& 1\end{array}\right]$

Hence it is clear that, ab has infinite order.