https://file.aiquickdraw.com/mathbot/file/1726762667784z1v6sdxl.jpg

The problem involves a transformation matrix and determining which specific transformation it represents. In the image, the matrix provided is:

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

This matrix can be analyzed as a transformation in three-dimensional space. The matrix shows that it swaps the \( x \)- and \( z \)-coordinates while keeping the \( y \)-coordinate unchanged. This corresponds to a **rotation through 90° about the \( y \)-axis**. Let's break down why:

- The first column of the matrix \((0, 0, 1)\) tells us that the new \( x \)-coordinate depends on the previous \( z \)-coordinate.
- The second column \((0, 1, 0)\) shows that the \( y \)-coordinate remains the same.
- The third column \((1, 0, 0)\) tells us that the new \( z \)-coordinate is now the old \( x \)-coordinate.

Thus, this is a **90° rotation about the \( y \)-axis**.

Would you like more details about this transformation or any part of the problem? 

Here are some follow-up questions:
1. What are the properties of rotations in 3D space?
2. How would a reflection matrix differ from a rotation matrix?
3. Can you identify how this transformation affects specific points in 3D space?
4. What would a rotation matrix around the \( x \)-axis look like?
5. How do eigenvalues relate to transformation matrices?

**Tip**: To quickly identify a transformation from a matrix, focus on how the columns modify the standard basis vectors.

The matrix \[
\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}
\] represents which of the following transformations?

Understand how the matrix \[
\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}
\] corresponds to a 90° rotation about the y-axis. This problem explores matrix transformations in 3D geometry, focusing on rotations and how matrix columns affect coordinate axes.

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