Last Updated on January 26, 2026 by Maged kamel
The easy approach to permutation matrix-part 2.
We will continue in Permutation matrix-part 2 to discuss the various types of permutation matrices for the 3×3 matrix.
The video that covers the content of this post is included from 12:32 to the end.
What is a permutation matrix P312?
This is the third arrangement of a 3×3 permutation matrix (P312), where there are swaps between rows 1 and 2, rows 2 and 3, and the last row, row 3, is moved to row 1.
Such a matrix, when multiplied by any other matrix, produces a new matrix in which the first row of the new matrix is the third row of the original matrix, and the second row of the new matrix is the first row of the original matrix. The third row of the new matrix is the second row of the original matrix.
An example of multiplying a permutation matrix P312 by a 3×3 matrix is shown in the next slide.
What is a permutation matrix P321?
In Permutation matrix-part 2, there is a fourth arrangement of a permutation matrix (3×3) which is called P321, where there is a swap between row 1 and row 3 and accordingly from row 3 to row 1, while the second row remains unchanged.
Such a matrix, when multiplied by any other matrix, produces a new matrix in which the first row is the third row of the original matrix and the second row is the same as the second row of the original matrix. The third row of the new matrix is the first row of the original matrix.
An example of multiplying a permutation matrix P321 by a 3×3 matrix is shown in the next slide.
What is a permutation matrix P132?
In Permutation matrix-part 2, there is a fifth arrangement of a permutation matrix (3×3), called P23, in which a swap occurs between rows 2 and 3, with row 3 then swapping with row 2, while the first row remains unchanged.
Such a matrix, when multiplied by any other matrix, produces a new matrix in which the first row is the same as the first row of the original matrix, and the second row is the third row of the original matrix. The third row of the new matrix is the second row of the original matrix.
A given example of multiplying a permutation matrix P132 by a 3×3 matrix is shown in the next slide image.
What is a permutation matrix P231?
In Permutation matrix-part 2, the last arrangement of a permutation matrix (3×3) is called P231, where there is a swap between row 1 and row 3, and also a swap from row 3 to row 2, and the second row r2 will be changed to row 1.
Such a matrix, when multiplied by any other matrix, produces a new matrix in which the first row of the new matrix is the second row of the original matrix, and the second row of the new matrix is the third row of the original matrix. The third row of the new matrix is the first row of the original matrix.
An example of multiplying a permutation matrix P231 by a 3×3 matrix is shown in the next slide.
The multiplication of a permutation matrix by its transpose.
The transpose of a permutation matrix is the inverse of that matrix. The next slide shows that multiplying P312 by its transpose yields the identity matrix. The details of the multiplication of P312 by its transpose are shown.
Here is an example of multiplying P132 by itself: its transpose yields the identity matrix.
The content of this post and the next post can be reviewed in the PDF file below.
This is a link to the first part- post -6- Easy introduction to permutation matrix.
This is a link to the following post: Solved Example for LU decomposition-partial pivoting
This is the Omni calculator for estimating various linear algebra items – LU Decomposition Calculator.
Link to Omni calculator-LU Decomposition Calculator.
Another calculator to use is the Calculator for matrices.