Math Problem Statement
Is the matrix below in echelon form, reduced echelon form, or neither?
Solution
Let's analyze the matrix:
1 & 0 & 4 & -5 \\ 0 & 1 & 5 & -5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Echelon Form: - A matrix is in **echelon form** if: 1. All nonzero rows are above rows of all zeros. 2. The leading entry of each nonzero row (called the pivot) is to the right of the leading entry of the row above it. 3. The leading entry in each row is 1. 4. All entries below the leading 1 in a column are zero. In this matrix: - The first two rows have leading 1s, and all entries below these leading 1s are zero. - All zero rows are at the bottom. Thus, **the matrix is in echelon form**. ### Reduced Echelon Form: - A matrix is in **reduced echelon form** if: 1. It is in echelon form. 2. The leading entry in each nonzero row is the only nonzero entry in its column. In this matrix: - The first row has a leading 1, but there are nonzero entries (4 and -5) in that row's column. - The second row has a leading 1, but the same column has a nonzero entry (5) in the third column. Thus, **the matrix is not in reduced echelon form**. ### Conclusion: The matrix is in **echelon form**, but **not in reduced echelon form**. Would you like more details on the steps involved or any specific questions? Here are 5 relative questions to deepen your understanding: 1. What is the difference between echelon form and reduced echelon form in terms of leading entries? 2. How would you transform this matrix into reduced row echelon form? 3. Why do matrices in reduced echelon form have unique solutions to linear systems? 4. Can you describe a method to identify the pivot positions in a matrix? 5. How does the echelon form help in solving systems of linear equations? **Tip:** When converting a matrix to reduced row echelon form, always make sure that each pivot is the only non-zero element in its column.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Echelon Form
Formulas
-
Theorems
Echelon Form Conditions
Reduced Row Echelon Form
Suitable Grade Level
College level or higher