Final answer:
The question involves finding orthogonal matrices that represent the symmetries of a square, which have both rotational and mirror symmetries. Eight orthogonal matrices corresponding to these symmetries are presented, with each matrix describing the transformation of points on the square under a particular symmetry operation.
Step-by-step explanation:
The student's question involves expressing the symmetries of a square using orthogonal matrices. A square has rotational and mirror symmetries that can be represented by such matrices, which are part of a subgroup isomorphic to D₄, the dihedral group of order 8.
For the rotational symmetries, we consider rotations by 0°, 90°, 180°, and 270° about the center of the square. Each of these can be represented by an orthogonal matrix:
- 0° rotation (identity transformation): The orthogonal matrix is I = \[
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\] - 90° rotation: The orthogonal matrix is R₉₀ = \[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\] - 180° rotation: The orthogonal matrix is R₁₈₀ = \[
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\] - 270° rotation: The orthogonal matrix is R₂⁷₀ = \[
\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
\]
For the mirror symmetries, these involve reflections over the lines y = x, y = -x, x = 0, and y = 0:
- Reflection over line y = x: The orthogonal matrix is Mₓ₀ = \[
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\] - Reflection over line y = -x: The orthogonal matrix is Mₓ₀₀ = \[
\begin{bmatrix}
0 & -1 \\
-1 & 0
\end{bmatrix}
\] - Reflection over the x-axis (y = 0): The orthogonal matrix is Mₓ₄ = \[
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
\] - Reflection over the y-axis (x = 0): The orthogonal matrix is Mₔ₂ = \[
\begin{bmatrix}
-1 & 0 \\
0 & 1
\end{bmatrix}
\]
Each orthogonal matrix provided corresponds to a specific symmetry operation of the square, and they all meet the conditions for being orthogonal (i.e., their columns are orthonormal and the matrix multiplied by its transpose yields the identity matrix).