In the above example how you find the permutation matrix P

To find a permutation matrix \( P \) that rearranges or permutes the rows or columns of another matrix, you follow these steps:

### Definition of a Permutation Matrix

A **permutation matrix** is a square binary matrix (a matrix where each entry is either 0 or 1) that has exactly one entry of 1 in each row and each column and 0s elsewhere. When a permutation matrix \( P \) is multiplied by another matrix, it permutes the rows (or columns) of the other matrix.

### Steps to Find the Permutation Matrix \( P \)

1. **Identify the Desired Permutation**: Determine the desired permutation of rows or columns. For example, if you have a matrix \( A \) and want to permute its rows or columns, you need to decide the new order of the rows or columns.

2. **Construct the Permutation Matrix**: Construct matrix \( P \) by placing 1s in positions that correspond to the desired row or column positions, and 0s elsewhere. 

   - For a row permutation matrix \( P \), multiplying it from the left with a matrix \( A \) (i.e., \( PA \)) permutes the rows of \( A \).
   - For a column permutation matrix \( P \), multiplying it from the right with a matrix \( A \) (i.e., \( AP \)) permutes the columns of \( A \).

3. **Form the Matrix \( P \) Step-by-Step**:
   - For example, if you have a 3x3 matrix and want to swap the first and second rows, the permutation matrix \( P \) would be:
   \[
   P = \begin{bmatrix}
   0 & 1 & 0 \\
   1 & 0 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]
   Here, the first row of \( P \) indicates that the new first row will be the second row of the identity matrix, the second row indicates that the new second row will be the first row, and the third row remains the same.

4. **Apply the Permutation Matrix**:
   - To permute the rows of a matrix \( A \), compute \( P \cdot A \).
   - To permute the columns of a matrix \( A \), compute \( A \cdot P \).

### Example

Suppose you have the following 3x3 matrix \( A \):

\[
A = \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\]

You want to rearrange \( A \) such that the new order of rows is 3, 1, 2 (i.e., the third row becomes the first, the first becomes the second, and the second becomes the third).

The corresponding permutation matrix \( P \) will be:

\[
P = \begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{bmatrix}
\]

Now, if you multiply \( P \) with \( A \):

\[
PA = \begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
= \begin{bmatrix}
a_{31} & a_{32} & a_{33} \\
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{bmatrix}
\]

This results in the matrix \( A \) with its rows rearranged as desired.

### Summary

To find a permutation matrix:

- Determine the desired new order of rows or columns.
- Construct the matrix \( P \) with 1s corresponding to the new order and 0s elsewhere.
- Apply \( P \) to permute rows (left multiplication) or columns (right multiplication).

Would you like to see more examples or have any questions about permutation matrices?  

Here are some related questions to consider:

1. How do you construct a permutation matrix to permute columns instead of rows?
2. What is the effect of multiplying a permutation matrix by its transpose?
3. Can a permutation matrix have a determinant of -1?
4. How do permutation matrices relate to the concept of matrix inversion?
5. What is the role of permutation matrices in LU decomposition?

**Tip:** When constructing a permutation matrix, always ensure that there is exactly one 1 in each row and column to maintain its properties.

Discover how to find a permutation matrix P that rearranges rows or columns of another matrix in linear algebra. This guide explains the definition of permutation matrices, steps to construct P, and examples of their application. Suitable for advanced undergraduate students and anyone interested in matrix operations.

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