6b- Practice problems for permutation matrices

Last Updated on October 21, 2026 by Maged kamel

Practice problems for permutation matrices.

The Post has two parts: the first contains educational material on solving mathematical problems involving permutation matrices, specifically from Timothy Sauer’s Numerical Analysis textbook. These exercises show that multiplying a Matrix on the left by a permutation Matrix swaps rows, for example, exchanging the Second and fifth rows in a 5×5 Matrix. The examples further explore right-side multiplication, showing that this operation shifts column positions rather than 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 expresses a permutation Matrix as I-J. We will discuss this in the Second video.

The video covers the first part of this Post.

The video timestamps are shown.

• 00:00 : ; Intro: Introduction; a summary of the ncontent.
• 00:35 Practice Problem #5 for permutation matrices.
• 04:38 Practice Problem #6 for permutation matrices- part a.
• 07:42 Practice Problem #6 for permutation matrices- part b.
• 09:59 Practice Problem #7 for permutation matrices.

Practice problems for permutation matrices.

We will 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 a 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 is 1, 2, 3, 4, 5. Starting from the first Row, the first entry is 1; all other entries are 0. The Second Row starts as 0 1. The remaining elements are 0. In the third Row, the 1 is in the third column; all other elements are zeros. Row 4 is 00010.

In the final Row of the 5-by-5Matrixx, the 1 is in the fifth position. Please refer to the next slide.

Now we need to exchange rows 2 and 5. Row 1 remains as it is. Row 2 moves to the fifth position, so 2 becomes 5; the third Row remains, the fourth Row remains, and Row 5 becomes d r

So, guided by the identity Matrix arrangement. need to swap to the permutation Matrix changes from P12345 to P15342. Consider 5-by- Matrix to the right.

The permutation Matrix.

Matrix A is 5×5, and the 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 001 will move the last Row to the Second position. The third Row remains, the fourth Row remains, and the Second column will come to the end.

After multiplication, the Second Row becomes the fifth Row. Please refer to the highlighted Row positions after multiplying by P from the left.

Second Practice Problem -6-part a-for permutation Matrix.

We will move to the next exercise. Part A: Find a 4×4 Matrix B such that multiplying P on the left by B exchanges the Second and fourth rows of B. 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 4×4 identity Matrix is 1000-0100-0010-0001. Double-check: we have one 1, with all other entries zero in every Row and column.

Okay, this is a normal position. For part, 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 a permutation Matrix.

We keep Matrix A on the left and the permutation Matrix on the right.

The permutation Matrix exchanges the columns. As we know, the nonzeros column 1000 is the same.

It should be followed by (0 1 0 0); the Second column is (0 0 1 0); and the fourth column is (0 0 0 1). The position is 4. The permutation Matrix P1432 will have different column arrangements. ere will be C143 instead of C1234. The first column, 1, 91913, will 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 become column 4, while the fourth column becomes 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 a permutation Matrix.

In the last slide, Practice Problem #7, the example changes four entries in the leftmost Matrix. Make the Matrix equation correct. We can see here that 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. This is the previous first Row. We have permutation P4231. Please refer to the following slide.

You can review the content of this Post and the next Post in the PDF file below.

Express a permutation Matrix as I-J.

We can express a permutation Matrix in I-J notation. The following video explains the various types of permutation Matrices.

The video timestamps are shown below.

• 00:00 Introduction: The I-J Expression for a permutation Matrix.
• 01:35 Why you need to understand the I-J Expression for a permutation Matrix.
• 03:28 What are P21 and P31 permutation matrices?
• 07:15 What are P32 and P231 permutation matrices?
• 12:15 What is a P312 permutation Matrix?

You can download the PDF for the Second part using the button below.

This is a Link to the first part, Post 6: Easy Introduction to permutation Matrix.

This introduces the following Post: 7-A Solved Problem 9-5 LU decomposition.

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 Matrixx Calculators.