Last Updated on September 18, 2025 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 permutation matrix (3×3), it is called P312, where there is a swap between row 1 to row 2, and also a swap from row 2 to row 3, and the last row r3 will be changed to row 1.
Such a matrix, when multiplied by any other matrix, will create a new matrix where the first row of the new matrix is the third row in the original matrix and the new second row is the first row of the original matrix. The third row of the new matrix is the second row of the original matrix.
A given example of the multiplication of a permutation matrix P312 with a 3×3 matrix is shown in the next slide image.
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 to row 3 and accordingly from r3 to r1, while the second row remains unchanged.
Such a matrix, when multiplied by any other matrix, will create a new matrix where the first row of the new matrix is the third row in 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.
A given example of the multiplication of a permutation matrix P321 with a 3×3 matrix is shown in the next slide image.
What is a permutation matrix P132?
In Permutation matrix-part 2, there is a fifth arrangement of a permutation matrix (3×3) it is called P23, where there is a swap between row 2 to row 3 and accordingly from r3 to r2, while the first row remains unchanged.
Such a matrix, when multiplied by any other matrix, will create a new matrix where the first row of the new matrix is the same as the first row in 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 the multiplication of a permutation matrix P132 with a 3×3 matrix is shown in the next slide image.
What is a permutation matrix P231?
In Permutation matrix-part 2, there is the last arrangement of a permutation matrix (3×3) is called P231, where there is a swap between row 1 to 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, will create a new matrix where the first row of the new matrix is the second row in the original matrix and the new second row is the third row of the original matrix. The third row of the new matrix is the first row of the original matrix.
A given example of the multiplication of a permutation matrix P231 with a 3×3 matrix is shown in the next slide image.
The multiplication of a permutation matrix by its transpose.
The transpose of a permutation matrix is the inverse of that matrix. The next slide image shows that the operation of multiplying P312 by its transpose will create an 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 will yield an identity matrix.
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Link to Omni calculator-LU Decomposition Calculator.
Another calculator to use is Calculator for matrices.