https://file.aiquickdraw.com/mathbot/file/1726681370278m4lawchq.jpg

The image asks to find two matrices, \( P \) and \( Q \), that perform specific transformations on a vector. Let’s break it down.

### 1. Matrix \( P \):
You are asked to find the matrix \( P \) that multiplies a vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) to give \(\begin{bmatrix} y \\ z \\ x \end{bmatrix}\).

Matrix \( P \) is already provided in the image as:
\[
P = \begin{bmatrix} 
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 
\end{bmatrix}
\]
This matrix reorders the vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) by moving \( z \) to the first position, \( x \) to the last position, and keeping \( y \) in the middle.

### 2. Matrix \( Q \):
You are also asked to find the matrix \( Q \) that multiplies \(\begin{bmatrix} y \\ z \\ x \end{bmatrix}\) to give back \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\).

Matrix \( Q \) is also provided:
\[
Q = \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\]
This is the identity matrix, which leaves any vector unchanged. So, this matrix doesn't alter the vector at all.

Would you like to go into further detail or ask any questions?

### Follow-up Questions:
1. How does the matrix \( P \) transform other types of vectors?
2. What would happen if you multiplied \( P \) by itself (i.e., \( P^2 \))?
3. Can matrix \( P \) be inverted? If so, what is its inverse?
4. Is there a general rule for constructing a permutation matrix like \( P \)?
5. What would matrix \( Q \) do if it were not the identity matrix?

**Tip**: Permutation matrices like \( P \) are useful in reordering elements of vectors, and they always have inverses.

Find the matrix P that multiplies [x, y, z] to give [y, z, x], and find the matrix Q that multiplies [y, z, x] to give back [x, y, z].

Learn how to find matrices P and Q that permute the vector [x, y, z] to [y, z, x], and reverse the operation. This problem covers key concepts in matrix multiplication and permutation matrices. Suitable for Grades 11-12 or college students studying linear algebra.

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