Last Updated on July 31, 2026 by Maged kamel
Permutation matrix.
Our subject will be the introduction to the permutation matrix. A permutation matrix is an identity matrix with rows interchanged.
I have created a YouTube video that covers this post and the next post.
The Timestamps for the video is shown next. The content of this post included from Time 0:00 till time12:23.
Using this permutation matrix, we can create a new matrix that differs from the old one by rearranging its rows. This is a brief illustration of the matrix quoted from Prof. Mathew W Reid. The author explains that, with the change in the multiplication process, we can obtain either a row swap or a column swap.
First, we start with the identity matrix for a 2×2 matrix. In the identity matrix, we have 1s in the upper-left and lower-right corners, and zeros in the remaining corners. Multiplying an identity matrix by a matrix 2xc2 will not change the arrangement of rows or columns.
How to swap the first row in a 2×2 matrix?
We have only two options in the 2×2 matrix for swapping rows. The first option is to swap the first and second rows when we multiply a given matrix from the left by a matrix P.
The second option is to swap the second row with the first row in the new matrix. Moving the first row of an identity matrix to the second row and vice versa, then multiplying by another matrix, yields a new matrix with shifted rows, as shown in the next slide.
How to swap the first column in a 2×2 matrix?
If we want to change the column arrangement of a matrix, we can multiply it by a permutation matrix with first column (0, 1). Its second column is (1 0). This arrangement will produce a new matrix with (b a) in the first column and (d c) in the second column. The original matrix arrangement was (a b) in the first row and (C d) in the second row. Multiplying any matrix from the right by a matrix ( 0 1, 1 0) will create a swap in the columns.
How many permutation matrices are in a 3×3 matrix?
We have (3x2x1)=6 permutation matrices for the 3×3 matrix. 6 sketches showing these arrangements in the next slide image.
The first option is the standard identity matrix I, where row 1 is followed by row 2, and row 3 is last. There is an interesting remark about the matrix: it has only one 1 in any of its columns or rows. No more than 3(1s) in the case of a 3×3 matrix. Multiplying the identity matrix by any 3×3 matrix will not change the row order.
What is a permutation matrix P213?
This is the second arrangement of a 3×3 matrix (P213), with a swap between rows 1 and 2, and the third row remains unchanged.
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 first row of the original matrix.
An example of multiplying a matrix P213 by a 3×3 matrix is shown in the next slide.
The content of this post and the next post can be reviewed via the following PDF file.
This is a link to the next post, which will be post 6a- Permutation matrix-part 2.
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.