Math Problem Statement
Which of the following matrices are in the Reduced Row Echelon Form (RREF)?
Solution
To determine which of the given matrices are in Reduced Row Echelon Form (RREF), we need to verify if they satisfy the following conditions:
- Leading 1s: The leading nonzero entry in each row is a 1 (pivot).
- Zeros above and below each leading 1: Every column containing a leading 1 has zeros everywhere else.
- Row order: Rows containing only zeros (if any) are at the bottom.
- Row position: The leading 1 in a row is to the right of the leading 1 in the row above it.
Let’s analyze each matrix one by one:
Matrix (a)
0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix}$$ - The second row has a leading 1 in the third column, which is valid. - However, the first row's leading entry is in the second column, and it should be the first non-zero entry in a row for RREF. - **Not in RREF**. ### Matrix (b) $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix}$$ - The first row has a leading 1 in the first column, which is valid. - The second row has a leading 1 in the third column, and all other elements in its column are 0. - **This matrix is in RREF**. ### Matrix (c) $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix}$$ - The first row has a leading 1 in the first column. - The second row has a leading 1 in the second column. - The third row has a leading 1 in the third column. - All elements above and below the leading 1s are zero, and the leading 1 in each row is to the right of the leading 1 in the row above it. - **This matrix is in RREF**. ### Matrix (d) $$\begin{pmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 1 & 2 \\ \end{pmatrix}$$ - The first row has a leading 1 in the first column. - The second row has a leading 1 in the second column. - However, there is a non-zero entry (2) in the third column of the first row, and it is not reduced to zero below the leading 1 in the second row. - **Not in RREF**. ### Matrix (e) $$\begin{pmatrix} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix}$$ - The first row’s leading entry should be in the first column, but the second column has a nonzero value (2) before the first zero appears in the second row. - **Not in RREF**. ### Matrix (f) $$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}$$ - The first row has a leading 1, but there is a non-zero entry (2) in the second column above the second row's leading 1. - **Not in RREF**. ### Final Answer: - **(b) and (c) are in RREF**. Would you like further details on RREF, or have any questions? Here are 5 related questions: 1. How can you reduce a matrix to RREF using Gaussian elimination? 2. What is the difference between Row Echelon Form (REF) and RREF? 3. What is the significance of leading 1s in RREF? 4. How is RREF used in solving systems of linear equations? 5. Can every matrix be transformed into RREF? **Tip:** While performing Gaussian elimination, always aim to zero out entries both below and above each pivot to achieve RREF.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Formulas
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Theorems
Reduced Row Echelon Form Criteria
Suitable Grade Level
Undergraduate level or advanced high school