Is the Matrix in RREF? Analyze and Understand Row-Reduced Echelon Form

Math Problem Statement

IS THIS IN RREF

To determine if the given matrix is in Row-Reduced Echelon Form (RREF), we need to check the following conditions:

Leading 1 in Each Row: Each nonzero row must have a leading 1, and the leading 1 of a row must be to the right of the leading 1 in the row above it.
Columns of Leading 1s: The columns containing the leading 1s must have zeros everywhere else.
All-Zero Rows: Any rows consisting entirely of zeros must be at the bottom of the matrix.
Reduced Form: Each leading 1 must be the only nonzero entry in its column.

Analyzing the matrix from your image:

1 & \frac{1}{2} & \frac{1}{2} \\ 0 & 1 & \frac{1}{3} \\ 0 & 0 & 1 \end{bmatrix}$$ 1. The first row has a leading 1 in the first column. 2. The second row has a leading 1 in the second column, and it is to the right of the first row's leading 1. 3. The third row has a leading 1 in the third column, and it is to the right of the second row's leading 1. 4. Each leading 1's column contains only zeros elsewhere. 5. There are no all-zero rows. **Conclusion**: The matrix satisfies all the criteria for Row-Reduced Echelon Form (RREF). It is indeed in RREF. Would you like further explanation, or do you have any specific questions? Here are five related questions to consider: 1. What are the conditions that define an echelon form versus RREF? 2. How does RREF help solve systems of linear equations? 3. What operations are used to convert a matrix to RREF? 4. Can every matrix be converted to RREF? Why or why not? 5. How do fractions like $$\frac{1}{2}$$ and $$\frac{1}{3}$$ affect computational processes in RREF? **Tip**: When working with matrices in RREF, always double-check the leading ones and their columns to confirm compliance with the rules!

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Linear Algebra
Row-Reduced Echelon Form (RREF)
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Criteria for RREF: leading 1s, zeros in columns of leading 1s, and placement of zero rows

Row-Reduced Echelon Form criteria

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