Math Problem Statement
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Determine whether the following matrices are in echelon form, reduced echelon form, or not in echelon form.
Solution
Let's analyze the matrices one by one and determine whether they are in echelon form, reduced echelon form, or not in echelon form.
Definitions:
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Echelon Form:
- All rows with only zeros are at the bottom.
- The leading entry (pivot) of each non-zero row is to the right of the leading entry of the row above it.
- The pivot entries do not have to be 1, and entries below pivots must be 0.
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Reduced Echelon Form:
- Satisfies all the properties of echelon form.
- Each pivot is 1, and the column containing a pivot has all other entries as 0.
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Not in Echelon Form:
- Violates the above conditions.
Matrix (a):
1 & 0 & 0 & -7 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ - This matrix satisfies the conditions for **echelon form**: - The zero row is at the bottom. - Pivots (1st and 2nd rows) are in the correct positions. - It is **not in reduced echelon form**, as the 4th column of the first row has a non-zero value (-7). **Answer: Echelon Form.** --- ### Matrix (b): $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$ - This matrix satisfies the conditions for **reduced echelon form**: - All pivots are 1. - Each pivot column has zeros elsewhere. **Answer: Reduced Echelon Form.** --- ### Matrix (c): $$\begin{bmatrix} 0 & 1 & 1 & 0 \\ 0 & -4 & 0 & 0 \end{bmatrix}$$ - This matrix is **not in echelon form**, as the pivot of the 2nd row (-4 in column 2) is not to the right of the leading entry of the 1st row (1 in column 2). **Answer: Not in Echelon Form.** --- ### Matrix (d): $$\begin{bmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$ - This matrix satisfies the conditions for **echelon form**, but not for **reduced echelon form** because the last column of the first row has a non-zero value (4). **Answer: Echelon Form.** --- ### Final Answers: - (a): **Echelon Form** - (b): **Reduced Echelon Form** - (c): **Not in Echelon Form** - (d): **Echelon Form** --- Would you like me to elaborate on any step or concept? 😊Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Echelon Form
Reduced Echelon Form
Formulas
Properties of echelon form
Properties of reduced echelon form
Theorems
Echelon Form Definition
Reduced Echelon Form Definition
Suitable Grade Level
College Level (Linear Algebra Course)