Math Problem Statement
solve matrix
Which of the following matrices are in row echelon form?
Solution
To determine which of the given matrices are in row echelon form (REF), we need to review the definition of row echelon form. A matrix is in row echelon form if:
- Any rows consisting entirely of zeros are at the bottom of the matrix.
- The leading entry (the first non-zero number from the left, called a pivot) of each non-zero row is strictly to the right of the leading entry of the row above it.
- The leading entry of each non-zero row is 1.
- The column containing the leading 1 has zeros below it.
Let's examine each matrix one by one.
Matrix (i):
0 & 0 \\ 0 & 0 \end{bmatrix}$$ - This matrix only contains zero rows. - It satisfies the REF conditions because zero rows are at the bottom, though there are no pivots. **This is in row echelon form.** ### Matrix (ii): $$\begin{bmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & -1 \end{bmatrix}$$ - The leading entries (1s) in each row are to the right of the leading entries in the rows above. - The leading 1s have zeros below them. - This matrix satisfies all the conditions of REF. **This is in row echelon form.** ### Matrix (iii): $$\begin{bmatrix} 1 & 4 & 3 & 7 \\ 0 & 1 & 6 & 2 \\ 0 & 0 & 1 & 5 \end{bmatrix}$$ - The leading entries (1s) in each row are to the right of the leading entries in the rows above. - The leading 1s have zeros below them. - This matrix satisfies all the conditions of REF. **This is in row echelon form.** ### Matrix (iv): $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ - In the second row, the leading entry is not 1 (since there’s a "1" in the first column in both the first and second rows). - This matrix does not satisfy the REF conditions. **This is not in row echelon form.** ### Final Answer: Matrices **(i), (ii), and (iii)** are in row echelon form, but matrix **(iv)** is not. Would you like further details or have any other questions? --- Here are 5 related questions you could explore: 1. What is the difference between row echelon form and reduced row echelon form? 2. How can a matrix be transformed into row echelon form? 3. What are some real-world applications of matrix row echelon forms? 4. How do you determine the rank of a matrix using its row echelon form? 5. Can any matrix be transformed into a row echelon form, and how is it done? **Tip:** When solving systems of equations using matrices, transforming the matrix into row echelon form simplifies finding solutions systematically.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Row Echelon Form
Formulas
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Theorems
Row Echelon Form Criteria
Suitable Grade Level
Undergraduate (Linear Algebra)