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 1E from Chapter 1.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 1E
Chapter:
Problem:
In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form...
Step-by-Step Solution
Step 1
For the given augmented matrices, we have to check whether the given matrix is in reduced echelon form are only in echelon form.
Conditions for Echelon Form:
1. All nonzero rows should be above the zero rows (if any).
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry must be zero in each column.
Conditions for Reduced Echelon Form: An echelon matrix is reduced echelon if it satisfies:
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Step 2: (a) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&1\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Thus, the given matrix is in Reduced Echelon form.
Step 3: (b) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Thus, the given matrix is in Reduced Echelon form.
Step 4: (c) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\]1. All nonzero rows are not above the zero rows.
Thus, the given matrix is in not in Echelon or reduced Echelon form.
Step 5: (d) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&1&0&1&1\\0&2&0&2&2\\0&0&0&3&3\\0&0&0&0&4\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
Thus, the given matrix is in in Echelon form.4. The leading entry in each nonzero row is not 1.
Thus, the given matrix is in not in reduced Echelon form.
ANSWERS
(a) Reduced Echelon Form
(b) Reduced Echelon Form
(c) Not in Echelon Form
(d) Echelon Form
For the given augmented matrices, we have to check whether the given matrix is in reduced echelon form are only in echelon form.
Conditions for Echelon Form:
1. All nonzero rows should be above the zero rows (if any).
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry must be zero in each column.
Conditions for Reduced Echelon Form: An echelon matrix is reduced echelon if it satisfies:
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Step 2: (a) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&1\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Thus, the given matrix is in Reduced Echelon form.
Step 3: (b) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only non-zero entry in its column.
Thus, the given matrix is in Reduced Echelon form.
Step 4: (c) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\]1. All nonzero rows are not above the zero rows.
Thus, the given matrix is in not in Echelon or reduced Echelon form.
Step 5: (d) Given Matrix
\[\left[ {\begin{array}{*{20}{c}}1&1&0&1&1\\0&2&0&2&2\\0&0&0&3&3\\0&0&0&0&4\end{array}} \right]\]1. All nonzero rows are above the zero rows.
2. Leading entry of each row is in a column to the right of the leading entry the row above it.
3. All elements below the leading entry are zero in each column.
Thus, the given matrix is in in Echelon form.4. The leading entry in each nonzero row is not 1.
Thus, the given matrix is in not in reduced Echelon form.
ANSWERS
(a) Reduced Echelon Form
(b) Reduced Echelon Form
(c) Not in Echelon Form
(d) Echelon Form