Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 2E from Chapter 1.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 2E
Chapter:
Problem:
Determine which matrices are in reduced echelon form and which others are only in echelon form.a. b. c. d.
Step-by-Step Solution
Given Information
we are given with some rectangular matrices, we have to determine if the matrix is in echelon form or in row-reduced form A rectangular matrix is in echelon form if it has the following properties:
1. All non-zero rows should be above the zero rows.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zero.
Also, an echelon matrix is in reduced echelon form if it has following additional conditions:i. The leading entry in each non-zero row is 1.
ii. Each leading 1 is the only non-zero entry in its column.
Step-1: (a)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&1&0&1\\0&0&{\left[ 1 \right]}&1\\0&0&0&0\end{array}} \right]\]In the given matrix, in each non-zero row the leading entry is 1. Also, each leading 1 is the only non-zero entry in its column.Hence,
This matrix is in reduced echelon form.
Step-2: (b)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&1&0&0\\0&{\left[ 1 \right]}&1&0\\0&0&{\left[ 1 \right]}&1\end{array}} \right]\]In the given matrix, in each non-zero row the leading entry is 1. However, each leading 1 is not the only non-zero entry in its column. From property 3(ii) given matrix is not a reduced echelon form but, the matrix is only in echelon form. Hence,
This matrix is in echelon form.
Step-3: (c)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&0&0&0\\1&{\left[ 1 \right]}&0&0\\0&1&{\left[ 1 \right]}&0\\0&0&1&{\left[ 1 \right]}\end{array}} \right]\]All the entries in a column below a leading entry are not zero. Thus, the condition (3) is not satisfied by the given matrix. Therefore, the given matrix is not in echelon form. Hence it can not be in reduced echelon form also. Hence,
The given matrix is neither in reduced echelon form nor in echelon form.
Step-3: (d)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}0&1&1&1&1\\0&0&2&2&2\\0&0&0&0&3\\0&0&0&0&0\end{array}} \right]\]In the given matrix, there are some leading entries that are not 1, so the matrix can not be in reduced echelon form. However, it satisfies all three conditions of an echelon matrix.
Tte given matrix is in echelon form
we are given with some rectangular matrices, we have to determine if the matrix is in echelon form or in row-reduced form A rectangular matrix is in echelon form if it has the following properties:
1. All non-zero rows should be above the zero rows.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zero.
Also, an echelon matrix is in reduced echelon form if it has following additional conditions:i. The leading entry in each non-zero row is 1.
ii. Each leading 1 is the only non-zero entry in its column.
Step-1: (a)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&1&0&1\\0&0&{\left[ 1 \right]}&1\\0&0&0&0\end{array}} \right]\]In the given matrix, in each non-zero row the leading entry is 1. Also, each leading 1 is the only non-zero entry in its column.Hence,
This matrix is in reduced echelon form.
Step-2: (b)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&1&0&0\\0&{\left[ 1 \right]}&1&0\\0&0&{\left[ 1 \right]}&1\end{array}} \right]\]In the given matrix, in each non-zero row the leading entry is 1. However, each leading 1 is not the only non-zero entry in its column. From property 3(ii) given matrix is not a reduced echelon form but, the matrix is only in echelon form. Hence,
This matrix is in echelon form.
Step-3: (c)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}{\left[ 1 \right]}&0&0&0\\1&{\left[ 1 \right]}&0&0\\0&1&{\left[ 1 \right]}&0\\0&0&1&{\left[ 1 \right]}\end{array}} \right]\]All the entries in a column below a leading entry are not zero. Thus, the condition (3) is not satisfied by the given matrix. Therefore, the given matrix is not in echelon form. Hence it can not be in reduced echelon form also. Hence,
The given matrix is neither in reduced echelon form nor in echelon form.
Step-3: (d)
Given matrix:\[\left[ {\begin{array}{*{20}{c}}0&1&1&1&1\\0&0&2&2&2\\0&0&0&0&3\\0&0&0&0&0\end{array}} \right]\]In the given matrix, there are some leading entries that are not 1, so the matrix can not be in reduced echelon form. However, it satisfies all three conditions of an echelon matrix.
Tte given matrix is in echelon form