Math Problem Statement
Which of the following matrices are in row-reduced form? (Note: The dotted vertical line in each matrix should be a single vertical line.)
Solution
To determine which of the matrices are in row-reduced form, we need to evaluate whether they meet the conditions for a matrix to be in reduced row echelon form (RREF). These conditions are:
- Leading entries (pivot positions) in each row must be 1.
- Each leading 1 must be the only non-zero entry in its column.
- Each leading 1 must be to the right of any leading 1s in the rows above it.
- All rows consisting entirely of zeros (if any) must be at the bottom.
Let's analyze each matrix:
Matrix I:
1 & 0 & 0 & | & 9 \\ 0 & 1 & 0 & | & -8 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$$ - The leading entries in each row are 1. - Each leading 1 is the only non-zero entry in its column. - The leading 1s are in different columns, and each subsequent leading 1 is to the right of the previous row's leading 1. - There are no rows consisting entirely of zeros. **Conclusion**: Matrix I **is** in reduced row echelon form. --- ### Matrix II: $$\begin{bmatrix} 1 & 1 & 1 & | & 9 \\ 0 & 0 & 1 & | & 0 \end{bmatrix}$$ - The leading entry in the first row is 1, but the next two entries are non-zero (1s), which violates the condition that the leading 1 must be the only non-zero entry in its column. - The second row has a leading 1 in the third column, which is correct. **Conclusion**: Matrix II **is not** in reduced row echelon form due to the first row violating the second condition. --- ### Matrix III: $$\begin{bmatrix} 1 & 0 & 9 & | & -8 \\ 0 & 0 & 1 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$ - The leading entry in the first row is 1, but there is a 9 in the third column of that row, which violates the condition that the leading 1 must be the only non-zero entry in its column. - The second row has a leading 1, which is correct. - The third row is all zeros, which is acceptable. **Conclusion**: Matrix III **is not** in reduced row echelon form due to the 9 in the first row violating the second condition. --- ### Final Answer: - Matrix I **is** in reduced row echelon form. - Matrices II and III **are not** in reduced row echelon form. Would you like more details or further clarifications? Here are 5 related questions to expand on this topic: 1. What are the specific steps to convert a matrix to reduced row echelon form? 2. How is Gaussian elimination different from Gauss-Jordan elimination? 3. What are the real-world applications of reduced row echelon form in solving systems of equations? 4. Can a matrix have more than one row-reduced form? 5. How does the number of leading 1s in a matrix relate to the rank of the matrix? **Tip**: When solving systems of equations using matrices, reducing a matrix to row-reduced form makes it easier to identify the solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Reduced Row Echelon Form
Formulas
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Theorems
Reduced Row Echelon Form Conditions
Suitable Grade Level
College-level or advanced high school