- #1

mathmari

Gold Member

MHB

- 5,049

- 7

We have the matrices \begin{equation*}s:=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}, \ d:=\frac{1}{2}\begin{pmatrix}-1 & -\sqrt{3} \\ \sqrt{3} & -1\end{pmatrix}\end{equation*} and the points \begin{equation*}p:=\begin{pmatrix}2 \\ 0 \end{pmatrix}, \ q:=\begin{pmatrix}-1 \\ \sqrt{3} \end{pmatrix}, \ r:=\begin{pmatrix}-1 \\ -\sqrt{3} \end{pmatrix}\end{equation*}

I draw the points $p, q, r$ and calculate also the points $sp, sq, sr, dp, dq, dr$ and I noticed that $s$ is a reflection as for the $x$-axis and $d$ is a rotation of $\frac{2\pi}{3}$.

1. Consider $G:=\{d, d^2, d^3, sd, sd^2, sd^3\}\subseteq \mathbb{R}^{2\times 2}$ and show that $G$ is closed as for multiplication of matrices, i.e. $gh\in G$ for all $g,h\in G$.

2. Show that the elements of $G$ are invertible and for each $g\in G$ there is $g^{-1}\in G$.

3. What is the geometric interpretation of $G$ ?

4. Let $z=\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}$ and $H:=G\cup \{zg\mid g\in G\}$. Show that $H\subseteq \mathbb{R}^2$ is closed as for multiplication of matrices. Let's start with question 1. Do we have to consider all possible combinations of the elements of $G$ and show that their product is again in $G$ ?

For example, do we have to do the following?

\begin{align*}d\cdot d^2&=d^3\in G \\ d\cdot d^3&=d^4=4-\text{times rotation about }120^{\circ}=\text{ratotaion about }480^{\circ}=\text{rotation about }360^{\circ}+120^{\circ}=\text{rotation about }120^{\circ}\\ & =d\in G \\ d\cdot s\cdot d&=\text{rotation about }120^{\circ}\text{ then reflection about x-axis and then rotation about }120^{\circ}\\ & =\text{reflection abour ax-axis}=s\in G\end{align*}

Or is there is shorter way? :unsure:

At question 2 do we have to show that the matrices are invertible? Or do we have to consider what the matrices represent? :unsure: