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Let's examine each matrix to determine whether it is in **echelon form** or **reduced echelon form**, or if it is not in echelon form at all.

### Definitions:
1. **Echelon Form**: 
   - All zero rows (if any) are at the bottom of the matrix.
   - The leading entry (first non-zero number from the left) of each non-zero row is strictly to the right of the leading entry of the row above it.
   - The leading entry in each non-zero row must be 1.

2. **Reduced Echelon Form** (REF):
   - Must meet all the conditions of echelon form.
   - Additionally, each leading 1 must be the only non-zero entry in its column.

---

### Matrix (a):
\[
\begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 1 \\
0 & 0 & -8 \\
\end{bmatrix}
\]

- Leading entries: First row has 1 in column 1, second row has 1 in column 2, third row has -8 in column 3.
- All non-zero rows are above any zero rows (there are no zero rows here).
- The leading entries shift to the right as we move down the rows.

This matrix **is in echelon form**, but not reduced echelon form because the leading entry in the last row is not 1.

### Matrix (b):
\[
\begin{bmatrix}
3 & 1 & 1 \\
-1 & 5 & 4 \\
0 & 0 & 1 \\
\end{bmatrix}
\]

- Leading entries: First row has 3, second row has -1, third row has 1.
- The leading entry in the second row is not strictly to the right of the leading entry in the first row.
- The leading entries in rows 1 and 2 are not 1.

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

### Matrix (c):
\[
\begin{bmatrix}
0 & 1 & 0 & 7 \\
0 & 0 & 1 & -8 \\
\end{bmatrix}
\]

- Leading entries: First row has 1 in column 2, second row has 1 in column 3.
- All non-zero rows are above any zero rows.
- The leading entries shift to the right as we move down the rows.
- Each leading 1 is the only non-zero entry in its column.

This matrix **is in reduced echelon form**.

### Matrix (d):
\[
\begin{bmatrix}
5 & 1 & 0 \\
0 & 1 & -10 \\
0 & 0 & -6 \\
\end{bmatrix}
\]

- Leading entries: First row has 5, second row has 1 in column 2, third row has -6 in column 3.
- The leading entry in the first row is not 1.
- The leading entries shift to the right as we move down the rows.

This matrix **is in echelon form**, but not reduced echelon form because the leading entries in rows 1 and 3 are not 1.

---

### Final Determination:
- (a) **Echelon form**
- (b) **Not in echelon form**
- (c) **Reduced echelon form**
- (d) **Echelon form**

Would you like more details on any part of this explanation?

### Follow-up Questions:
1. What are the specific conditions for a matrix to be in reduced echelon form?
2. How can you perform row reduction to convert a matrix into reduced echelon form?
3. Can a matrix in echelon form have negative leading entries?
4. How do the row operations (swap, multiply, and add) affect the echelon form?
5. What are the benefits of converting a matrix into reduced echelon form?

### Tip:
When working with matrices, it's helpful to clearly identify leading entries as you perform row operations to ensure you're meeting the conditions for echelon or reduced echelon form.

Determine whether the following matrices are in echelon form, reduced echelon form, or not in echelon form.

Determine the echelon or reduced echelon form for several matrices. Learn how to check whether matrices meet the conditions of echelon or reduced echelon form using row operations and leading entries. Suitable for Grades 10-12.

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