what matrix represents a 180 degree rotation clockwise

A 180-degree rotation clockwise (or counterclockwise, since a 180-degree rotation in either direction results in the same transformation) around the origin in 2D can be represented by the following matrix:

\[
\begin{pmatrix}
-1 & 0 \\
0 & -1
\end{pmatrix}
\]

This matrix reflects the transformation that negates both the \(x\) and \(y\) coordinates of any point, effectively rotating it 180 degrees about the origin.

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### Relative Questions:
1. How does a 90-degree rotation clockwise differ from a 180-degree rotation?
2. What would the matrix look like for a 270-degree rotation clockwise?
3. How does the determinant of the rotation matrix affect the area of a shape?
4. What are the new coordinates of a point \((x, y)\) after a 180-degree rotation?
5. Can a rotation matrix ever change the size of the shape it is transforming?

### Tip:
For any angle \(\theta\), the general rotation matrix is given by \( \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \). For specific angles like 90°, 180°, and 270°, substitute the angle into the general formula to get the corresponding rotation matrix.

Explore how a 180-degree rotation clockwise is represented by a specific matrix in 2D. Understand the mathematical transformation that negates both the x and y coordinates, providing a clear visualization of rotational symmetry. Suitable for advanced high school students and anyone interested in linear algebra concepts.

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