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Chapter 3: Problem 20

Write the given permutation matrix as a product of elementary (row interchange) matrices. $$\left[\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$

Short Answer

Expert verified
The permutation matrix is a product of \(E_{14} \cdot E_{23}\).

Step by step solution

01

Understanding the Permutation Matrix

The given permutation matrix is \[\begin{array}{llll}0 & 0 & 0 & 1 \0 & 0 & 1 & 0 \0 & 1 & 0 & 0 \1 & 0 & 0 & 0 \\end{array}\]This matrix is a permutation of the identity matrix, which indicates that rows have been interchanged. Our goal is to express this as a product of matrices that perform these row swaps.
02

Identifying Row Interchanges

Starting with an identity matrix \[\begin{array}{llll}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \\end{array}\]We need to reach the given permutation matrix. Analyze the permutation matrix:- First, swap row 1 with row 4.- Then, swap the new row 2 (original row 4) with row 3.
03

Representing Row Swaps with Elementary Matrices

Each row interchange is represented by an elementary matrix. The swap between rows 1 and 4 can be achieved by multiplying by:\[E_{14} = \begin{array}{llll}0 & 0 & 0 & 1 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \1 & 0 & 0 & 0 \\end{array}\]The swap between rows 2 and 3 can be achieved by:\[E_{23} = \begin{array}{llll}1 & 0 & 0 & 0 \0 & 0 & 1 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & 1 \\end{array}\]
04

Verifying the Product

Now, let's multiply the elementary matrices to verify the permutation matrix:- First, multiply the identity matrix by \(E_{14}\):\[\begin{array}{cccc}0 & 0 & 0 & 1 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \1 & 0 & 0 & 0 \\end{array}\]- Next, multiply the result by \(E_{23}\):\[\begin{array}{cccc}0 & 0 & 0 & 1 \0 & 0 & 1 & 0 \0 & 1 & 0 & 0 \1 & 0 & 0 & 0 \\end{array}\]This confirms that the product of these two elementary matrices yields the original permutation matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elementary Matrices
Elementary matrices are a key concept in linear algebra. An elementary matrix is derived from the identity matrix by performing a single elementary row operation. This concept is vital because any matrix can be transformed into simpler forms by using these row operations in sequence.
There are three kinds of elementary row operations:
  • Row swapping: Exchanging two rows in a matrix.
  • Scaling: Multiplying all elements of a row by a non-zero scalar.
  • Row addition: Adding a multiple of one row to another row.
For the permutation matrix question you are studying, we focus on the first type鈥攔ow swapping. To express a permutation matrix as a product of elementary matrices, row interchanges become the building blocks.
Row Interchange
Row interchange is simply swapping two rows in a matrix. This is one of the fundamental operations in linear algebra, and it greatly influences the arrangement of a matrix's rows. When presented in matrix form, each swap follows a pattern. For example, if you swap row 1 with row 4, it is represented by an elementary matrix where the first and fourth rows of the identity matrix have switched places.
Row interchanges are reversible, meaning if you swap the same rows again, you'll return to your original matrix. Thus, understanding which rows need to be interchanged helps us write a permutation matrix using elementary matrices.
Identity Matrix
The identity matrix is a special kind of square matrix. In it, all the diagonal elements are 1, while the rest are 0. This matrix acts as the "1" in matrix algebra, just as the number 1 is the multiplicative identity in arithmetic.
An important property of the identity matrix is that multiplying any matrix by it leaves the original matrix unchanged. This property makes the identity matrix a crucial starting point for transformations using elementary matrices. In our exercise, we started with the identity matrix and applied row swaps to arrive at the permutation matrix of interest.
In practical applications, using the identity matrix as a base form allows you to systematically apply and understand changes through elementary matrices.

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