Last Updated on October 21, 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 3×3 permutation matrices.
The video covering this Post is included from 12:23 to the end.
The video timestamps are shown from 12:23 to the end.
- 12:23 Permutation Matrix P312.
- 13:35 Permutation Matrix P321.
- 14:38 Permutation Matrix P132.
- 15:40 Permutation Matrix P231.
- 16:23 Property of a permutation Matrix.
What is a permutation Matrix P312?
This is the third arrangement of a 3×3 permutation Matrix (P312), with swaps between rows 1 and 2, rows 2 and 3, and the last Row (Row 3) moved to Row 1.
When you multiply such a Matrix by any other Matrix, the first Row of the result is the third Row of the original Matrix, and the Second Row is the first Row. 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 on 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) called P321, which swaps Row 1 and Row 3, while the Second Row remains unchanged.
When multiplied by any other Matrix, this 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 rows 2 and 3 are swapped, while the first Row remains unchanged.
When multiplied by any other Matrix, it 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 original Matrix’s Second Row.
An example of multiplying a permutation Matrix P132 by a 3×3 Matrix is shown on the next slide.
What is a permutation Matrix P231?
In Permutation matrix-part 2, the last arrangement of a permutation Matrix (3×3) is called P231, where Row 1 and Row 3 are swapped, and Row 2 is swapped with Row 3, so the Second Row r2 becomes the first Row.
When multiplied by any other Matrix, this Matrix produces a new Matrix where the first Row is the Second 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 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.
Multiplying a permutation Matrix by its transpose.
The transpose of a permutation Matrix is its Inverse. The next slide shows that multiplying P312 by its transpose yields the identity Matrix. The slide shows the details of multiplying P312 by its transpose.
Here is an example of multiplying P132 by itself: it yields the identity matrix.
You can review the content of this Post and the Previous Post in the PDF file below.
This Link link to the first part- Post 6:Introductionuction to permutation matri.ces
There is a Post with practice problems on permutation matrices that covers the IJ notation for permutation matrices.
This is the Omni calculator for estimating various linear algebra items: LU Decomposition Calculator.
Link to the Omni calculator: LU Decomposition Calculator.
Another calculator to use is the Calculator for matrices.