Last Updated on July 31, 2026 by Maged kamel
- Practice problems for permutation matrices.
- Practice problems for permutation matrices.
- First Practice problem -5 for permutation matrix.
- Second Practice problem -6-part a-for permutation matrix.
- Second Practice problem-6-part b-for permutation matrix.
- Review the second column after multiplying the permutation matrix from the right.
- Practice problem-7 for permutation matrix.
- Express permutation matrix as I-J.
Practice problems for permutation matrices.
the post includes two parts, the first part contains educational materials focused on solving mathematical problems involving permutation matrices, specifically from Timothy Sauer’s Numerical Analysis textbook. These exercises demonstrate how multiplying a matrix on the left by a specific permutation matrix swaps rows, such as exchanging the second and fifth rows in a 5×5 matrix. The examples further explore the mechanics of right-side multiplication, illustrating how this operation shifts the positions of columns instead of rows.
Through step-by-step visual demonstrations, the source clarifies how to construct these matrices by modifying the identity matrix to achieve desired structural changes. Overall, the content serves as a practical guide for understanding linear algebra transformations through worked-out exercise solutions.
The second part contains the expreession of permutation matrix as I-J.There will be a discussion expressed in the second video.
The video covers the content of this post for the first part.
The Timestamps for the video are shown.
Practice problems for permutation matrices.
We are going to look at practice problems on permutation matrices. Quoted from the book Numerical Analysis by Professor Timothy Sauer. These Practice problems are 5, 6 and 7 from Chapter 2.4. Exercise.
The data for these practice problems are shown on the next slide.
First Practice problem -5 for permutation matrix.
Write down a 5-by-5 matrix P such that multiplication of another matrix by P on the left causes row two and five to be exchanged.
Okay, what is the normal position? Please refer to the following slide on the bottom left side of the slide,
The normal position of the identity matrix 1, 2, 3, 4, 5. Starting from the first row, the first entry is 1; all other entries are 0. For the second row, it starts as 0 1. The remaining elements are 0. For the third row, the 1 is in the third column; all other elements are zeros. And row 4 is expressed as 00010.
And in the final row of the 5-by-5 matrix, the 1 occupies the fifth position. Please refer to the next slide.
It is now required to exchange rows 2 and 5. So 1 remains as it is. Row 2 will move to the fifth position, or 2 will become 5; the third row remains, the fourth row remains, and row 5 becomes 2nd row.
So, guided by the arrangement of the identity matrix. I need to swap to 5. The permutation matrix changes from P12345 to P15342. Let us consider the 5-by-5 matrix to the right.
The permutation matrix.
Matrix A is 5×5, and rows are a,b,c,d,e,f,g, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y.
Once we have 1 0 0 0 in the identity matrix, it is the same.
So we expect that ABCDE to remain as it is. Placing 00001 will bring the last row to the second position. The third row remains, the fourth row remains, and the second column will come to the end.
The second row will become the fifth row after multiplication. Please refer to the highlighted row positions after multiplying with P from the left side.
Second Practice problem -6-part a-for permutation matrix.
We will move to the next exercise. Number 6. Write down a 4 by 4 matrix B such that multiplying the matrix on the left by P causes the second and fourth rows of the matrix to be exchanged. This is part A. For Part B. What is the effect of multiplying P on the right? Demonstrate with an example.
We have a 4×4 matrix, and the identity matrix for 4×4 is 1000-0100-0010-0001. To double-check, we have only one 1, with all the remaining entries zero across all columns and rows.
Okay, this is a normal position. For part A, we need to multiply the 4×4 matrix by the P1423 matrix, where the first row is 10000, the 2nd row is modified to 0001, the third row is 0010 unchanged, and the last row is 0010 after the change. After multiplication, the first and third rows remain unchanged, while the second and fourth rows swap.
Second Practice problem-6-part b-for permutation matrix.
We keep matrix A on the left and the permutation matrix on the right.
What happened is that the permutation matrix will exchange the columns. As we know, the normal column 1000 is the same.
It should be followed by (01 0 0); the second column is 0 0 1 0; and the fourth column is (0 0 0 1). The normal position is C1234. The permutation matrix P1432 will have different column arrangements. There will be C1432) instead of C1234. The first column 15913will be unchanged. Please refer to the following slide for more details.
An example of multiplying a permutation matrix P231 by a 3×3 matrix is shown in the next slide.
Review the second column after multiplying the permutation matrix from the right.
The second column will be shifted to be column 4, while the fourth column is becoming the new column, so we have as a new second column 4 8 12 16)
The following slide shows the arrangement of the three columns.
The following slide shows the arrangement of the final four columns.
Practice problem-7 for permutation matrix.
In the last slide, Practice problem #7, the example changes four entries in the leftmost matrix. To make the matrix equation correct. We can see here we have (1, 2, 3, 4), (3, 4, 5, 6), (5, 6, 7, 8) and (7, 8, 9, 0).
It is multiplied by a permutation matrix such that the right-hand side will become (5, 6, 7,8), (3, 4, 5, 6), (7, 8, 9, 0). and (1,2,3,4)).
We can see that starting in the new matrix.
After multiplication by the permutation matrix, we have the third row, then the second, then the fourth. And this last row is the previous first row. We have used the permutation matrix P4231. Please refer to the following slide.
The content of this post and the next post can be reviewed in the PDF file below.
Express permutation matrix as I-J.
We can express permution matrix as I-J notation.The following video explains the various types of the permutation Matrices.
The Timestamps for the video are shown.
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.