Math Problem Statement
Determine whether the following matrices are in echelon form, reduced echelon form, or not in echelon form.
Solution
Let's analyze each matrix from the uploaded image to determine whether they are in echelon form, reduced echelon form, or neither.
Conditions for Row Echelon Form (REF):
- All zero rows (if any) are at the bottom.
- The leading entry (pivot) of each nonzero row is to the right of the leading entry of the row above it.
- Each pivot must be a nonzero value.
Conditions for Reduced Row Echelon Form (RREF):
- The matrix is in row echelon form.
- Each pivot is 1 (normalized).
- Each pivot is the only nonzero entry in its column.
Analyzing the matrices:
a.
Matrix:
1 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 6 \end{bmatrix}$$ - Row 2 is entirely zero, which is acceptable for echelon form. - However, the leading entry in Row 3 (column 2) is not to the right of the leading entry in Row 1 (column 1). **Conclusion**: **Not in Echelon Form**. --- #### **b.** Matrix: $$\begin{bmatrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & -6 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$ - All zero rows (if any) are at the bottom. - The pivots are in columns 1, 2, and 3 and are positioned to the right of the pivot above. - The pivots are 1, and there are no nonzero entries above each pivot in its column. **Conclusion**: **Reduced Row Echelon Form**. --- #### **c.** Matrix: $$\begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$$ - Row 2 is entirely zero, which is fine for echelon form. - However, the leading entry in Row 1 is in the first column, and Row 2 lacks a pivot in the second column, violating the pivot rule. **Conclusion**: **Not in Echelon Form**. --- #### **d.** Matrix: $$\begin{bmatrix} 10 & 7 & 9 & 2 & 3 \\ 0 & 0 & 1 & 1 & 10 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$ - All zero rows (Row 4) are at the bottom. - The leading entries (pivots) in Row 2 and Row 3 are in columns 3 and 4, respectively, and each is to the right of the pivot above. - The first row does not meet echelon form rules because the leading entry (pivot) is not normalized to 1. **Conclusion**: **Not in Echelon Form**. --- ### Final Results: - **a:** Not in Echelon Form - **b:** Reduced Row Echelon Form - **c:** Not in Echelon Form - **d:** Not in Echelon Form --- Would you like me to provide detailed explanations for each condition? --- ### 5 Related Questions: 1. What are the key differences between echelon form and reduced echelon form? 2. How do you convert a matrix into row echelon form? 3. Can a matrix have multiple reduced row echelon forms? 4. What role do pivots play in linear algebra and solving systems of equations? 5. Why must the pivots always be normalized in reduced echelon form? ### Tip: To quickly check if a matrix is in reduced echelon form, ensure that each pivot is 1 and all other entries in the pivot column are 0!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Row Echelon Form
Reduced Row Echelon Form
Formulas
Conditions for Echelon Form
Conditions for Reduced Echelon Form
Theorems
Row Echelon Form
Reduced Row Echelon Form
Suitable Grade Level
Grades 11-12