Closure under matrix multiplication for 2x2 matrices

Question

I need to show that this set is closed under matrix multiplication, is there a better way than doing it via a Cayley table? Or rather I assume there is and I just can't get my head around it. Any help would be greatly appreciated.

$\begin{gather*} M= \left\{ \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \right\}\\ \end{gather*}$

Answer 1

Call matrices of this $\begin{pmatrix}*&0\\0&*\end{pmatrix}$ form "diagonal" and matrices of this $\begin{pmatrix}0&*\\*&0\end{pmatrix}$ form "skew diagonal". First prove that the product of diagonal matrices is diagonal, the product of diagonal and skew diagonal is skew diagonal, the product of skew diagonal and diagonal is skew diagonal and that the product of skew diagonal and skew diagonal is diagonal.

Then prove that every entry in the resulting matrices will always be $\pm 1$. Finally note that every matrix which is diagonal or skew diagonal and has entries in $\{\pm 1\}$ is one of your eight.

Answer 2

One way would be to demonstrate a group isomorphism between your set and a group where you know the operation is closed.

If you're familiar with the dihedral group of order 8, that might be a good place to start. Then note that as the dihedral group is generated by the basic operations of rotations and flips, so it would be sufficient to show that the group isomorphism works for those two elements mapped to $R$ and $F$.

Answer 3

A first step is to determine the order of the $8$ elements. We can easily see that not all matrices commute, so that we have two distinct non-abelian groups of order $8$, which may be represented by these matrices, namely the dihedral group $D_4$ or the quaternion group $Q_8$. Now the information of the order of elements is helpful, because we see that it represents the dihedral group - which is closed under multiplication. Also, here we see that this is correct.

Answer 4

Here's a second way to do it which might be quicker. Define $G$ to be the group of the $2\times 2$ invertible matrices which fix the set $\{(0,1),(1,0),(0,-1),(-1,0)\}$. It's easy to prove this is a group and also easy to check that it contains precisely your eight matrices.

Answer 5

Your group $M$ can be described

$$M = \left\lbrace \begin{pmatrix} a_1 &0\\ 0 &b_1\end{pmatrix}, \begin{pmatrix} 0 &a_2\\ b_2 &0\end{pmatrix} : a_i, b_i \in (\Bbb Z/3\Bbb Z)^\times\right\rbrace.$$

Then you need only note that any product $a_1a_2$ or $b_1b_2$ lands back in $(\Bbb Z/3\Bbb Z)^\times$ as this is a group.

Answer 6

The matrices $A = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$ and $B = \begin{bmatrix}0&1\\1&0\end{bmatrix}$ satisfy the identities $BB=I$, $AA = I$, $AB=-BA.$ Consider the group of all possible matrix products of $A$s and $B$s, consider any element of it. The last identity allows you to reorder any matrix multiplication of these two into the form $\pm A\dots AB\dots B$ and the first two identities allow you to remove As and Bs from it two-at-a-time, so the group has at most 8 elements, $$\{I, A, B, AB, -I, -A, -B, -AB\}.$$ By inspection, these are the 8 that you have and none of them is secretly equal to another of them, so the group has order 8.

Answer 7

This is not a rigorous proof, but an interesting geometric intuition. Those matrices are different kinds of reflections (about origin, about $y=x$, etc). As an example, pick $(2,1)$ and transform it with each of those matrix. You have (label $i$ means the $i$th matrix is applied)

It's apparent that those transformations are "geometrically closed".

Answer 8

Another geometric way: Fix the standard orthonormal frame. Then the given diagonal matrices each either fix or reflect the basis vectors, e.g. $\left(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right)$ fixes the first and reflects the second. The matrix $\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0\end{smallmatrix}\right)$ is a change of orientation and all skew diagonal ones are given by this change of orientation, e.g. $\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0\end{smallmatrix}\right)=\left(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right)\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0\end{smallmatrix}\right)$. We may assign a triple $(a,b,c)$ from $\{-1,1\}\times \{-1,1\}\times \{-1,1\}$ to each matrix, where $a$ indicates whether the first basis vector is reflected or not, $b$ for the second and $c$ tracks whether we had a change of orientation. Hence we see that we can identify your matrices with the group $\mathbb{Z}_2^3$ which is closed.