https://file.aiquickdraw.com/mathbot/file/1726455551865cmiieucr.jpg

which matrices are in reduced
echelon form and which others are only in echelon form.

The matrices in the image are as follows:

- Matrix (a): \(\begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}\)
- Matrix (b): \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)
- Matrix (c): \(\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}\)
- Matrix (d): \(\begin{bmatrix} 0 & 1 & 1 & 1 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 3 \end{bmatrix}\)

### Analysis of Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)

1. **Matrix (a)**:
   - **Form**: Echelon Form (REF)
   - **Reason**: Each leading coefficient (pivot) is 1, and each is to the right of the pivot above it. However, the first row has a non-zero entry in the same column as a pivot below it, which does not meet the criteria for RREF.
   
2. **Matrix (b)**:
   - **Form**: Reduced Row Echelon Form (RREF)
   - **Reason**: Every leading coefficient is 1, and each is the only non-zero entry in its column, which meets the criteria for RREF.
   
3. **Matrix (c)**:
   - **Form**: Neither REF nor RREF
   - **Reason**: The first and second rows are not ordered correctly by leading coefficients (the second row should be all zeros since it duplicates the first), and the third row does not have the leading coefficient to the right of the pivot in the second row.
   
4. **Matrix (d)**:
   - **Form**: Echelon Form (REF)
   - **Reason**: Each leading coefficient is the only non-zero entry in its row up to that point, and each subsequent leading coefficient is to the right of the one above it. However, it is not in RREF because the leading coefficients are not 1, and there are non-zero entries above and below the pivots.

If you have any questions or need further details on this topic, feel free to ask! Here are some additional questions you might consider:

1. What are the steps to convert a matrix to RREF?
2. Why is matrix (c) neither REF nor RREF?
3. How can matrix (d) be further simplified to reach RREF?
4. What are the practical applications of using RREF in solving systems of linear equations?
5. How does the presence of a row of zeros affect the rank of a matrix?

**Tip**: Understanding the structure and properties of different matrix forms can significantly help in solving systems of linear equations and understanding vector spaces.

Which matrices are in reduced echelon form and which others are only in echelon form?

Analyze which matrices are in echelon form and reduced row echelon form from a set of given matrices. This solution explains the criteria for echelon and reduced echelon forms, making it ideal for high school algebra students.

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