Klein Four-Group as Order 2 Matrices

Theorem
Let $G$ be the set of order $2$ square matrices:


 * $G = \left\{{I, A, B, C}\right\}$

where:
 * $I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, \quad A = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, \quad B = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix}, \quad C = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$

Then the algebraic structure $\left({G, \times}\right)$, where $\times$ denotes (conventional) matrix multiplication, forms the Klein four-group.

Proof
From Unit Matrix is Unity of Ring of Square Matrices, $I$ can be identified as the unit matrix of order $2$.

Then:
 * $A^2 = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I$
 * $AB = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} = C$
 * $AC = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} = B$


 * $B^2 = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I$
 * $BA = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} = C$
 * $BC = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} = A$


 * $C^2 = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I$
 * $CA = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} = B$
 * $CB = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix} = A$

Putting this together into a Cayley table:


 * $\begin{array}{c|cccccc}

& I & A & B & C \\ \hline I & I & A & B & C \\ A & A & I & C & B \\ B & B & C & I & A \\ C & C & B & A & I \\ \end{array}$

it is verified by inspection that this is an instance of the Klein four-group.