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are the matrices in row echelon form, reduced row echelon form or not in row echelon form

Let's go through the matrices one by one to identify if they are in **Row Echelon Form (REF)** or **Reduced Row Echelon Form (RREF)**.

### Matrix 1:
\[
\begin{bmatrix}
1 & 0 & 0 & -3 \\
0 & 1 & 0 & -8 \\
0 & 0 & 1 & 8
\end{bmatrix}
\]
- This matrix has leading 1's in each row.
- Each leading 1 is the only non-zero entry in its column.
- All rows with non-zero elements are above any rows of all zeroes (there are no zero rows here).
- Every leading 1 is to the right of the leading 1 in the row above it.

This matrix satisfies all conditions for **RREF**.

### Matrix 2:
\[
\begin{bmatrix}
0 & 1 & 0 & -7 \\
0 & 0 & 1 & -4
\end{bmatrix}
\]
- This matrix does **not** have a leading 1 in the first column (the first non-zero element in the first row is in the second column).
- The second row is fine, with a leading 1 in the third column.

This matrix is **not in row echelon form** because the leading 1 must appear earlier (in the first column if possible).

### Matrix 3:
\[
\begin{bmatrix}
1 & 0 & 8 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 4 \\
0 & 0 & 0 & 1 & 0
\end{bmatrix}
\]
- There are leading 1's in each row.
- Each leading 1 is to the right of the leading 1 in the row above it.
- All rows of zeroes (none here) would be at the bottom.

This matrix satisfies **Row Echelon Form (REF)**. However, it is **not in RREF** because there are non-zero entries (like 8 and 4) above the leading 1's.

### Matrix 4:
\[
\begin{bmatrix}
2 & 1 & 1 \\
1 & -6 & 8 \\
0 & 0 & 1
\end{bmatrix}
\]
- The first row has a 2 in the first position, which should be a 1 for REF or RREF.
- There are no leading 1's properly positioned.

This matrix is **not in row echelon form**.

### Matrix 5:
\[
\begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 1 \\
0 & 0 & -6
\end{bmatrix}
\]
- Leading 1's are properly positioned in the first and second rows.
- However, the third row does not have a leading 1; instead, it has a -6.

This matrix is in **Row Echelon Form (REF)**, but not in RREF since the third row is not normalized to a leading 1.

### Matrix 6:
\[
\begin{bmatrix}
-10 & 1 & 1 & 0 \\
0 & 1 & -10 & 0 \\
0 & 0 & 0 & 10
\end{bmatrix}
\]
- The first row should have a leading 1 instead of -10.
- The third row has a leading 10, which should be normalized to 1.

This matrix is **not in row echelon form**.

---

In summary:
1. **Matrix 1**: RREF
2. **Matrix 2**: Not in row echelon form
3. **Matrix 3**: REF
4. **Matrix 4**: Not in row echelon form
5. **Matrix 5**: REF
6. **Matrix 6**: Not in row echelon form

---

Would you like any further details or have any questions?

Here are 5 questions to explore further:
1. What are the key differences between Row Echelon Form and Reduced Row Echelon Form?
2. How can you transform a matrix into its RREF?
3. Why is it important to have leading 1's in REF or RREF?
4. How does Gaussian elimination relate to finding the row echelon form?
5. Can all matrices be transformed into RREF?

**Tip**: To quickly verify if a matrix is in RREF, check if all leading 1’s are isolated in their columns and if each leading 1 is to the right of the leading 1 in the row above.

Are the matrices in row echelon form, reduced row echelon form, or not in row echelon form?

This page explores six matrices, determining whether each is in row echelon form, reduced row echelon form, or neither. It covers core concepts in linear algebra and matrix transformations, making it suitable for advanced high school or college-level students.

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