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To determine which of the matrices are in row-reduced form, we need to evaluate whether they meet the conditions for a matrix to be in *reduced row echelon form* (RREF). These conditions are:

1. **Leading entries (pivot positions) in each row must be 1.**
2. **Each leading 1 must be the only non-zero entry in its column.**
3. **Each leading 1 must be to the right of any leading 1s in the rows above it.**
4. **All rows consisting entirely of zeros (if any) must be at the bottom.**

Let's analyze each matrix:

### Matrix I:
\[
\begin{bmatrix}
1 & 0 & 0 & | & 9 \\
0 & 1 & 0 & | & -8 \\
0 & 0 & 1 & | & 3
\end{bmatrix}
\]
- The leading entries in each row are 1.
- Each leading 1 is the only non-zero entry in its column.
- The leading 1s are in different columns, and each subsequent leading 1 is to the right of the previous row's leading 1.
- There are no rows consisting entirely of zeros.

**Conclusion**: Matrix I **is** in reduced row echelon form.

---

### Matrix II:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 9 \\
0 & 0 & 1 & | & 0
\end{bmatrix}
\]
- The leading entry in the first row is 1, but the next two entries are non-zero (1s), which violates the condition that the leading 1 must be the only non-zero entry in its column.
- The second row has a leading 1 in the third column, which is correct.

**Conclusion**: Matrix II **is not** in reduced row echelon form due to the first row violating the second condition.

---

### Matrix III:
\[
\begin{bmatrix}
1 & 0 & 9 & | & -8 \\
0 & 0 & 1 & | & 3 \\
0 & 0 & 0 & | & 0
\end{bmatrix}
\]
- The leading entry in the first row is 1, but there is a 9 in the third column of that row, which violates the condition that the leading 1 must be the only non-zero entry in its column.
- The second row has a leading 1, which is correct.
- The third row is all zeros, which is acceptable.

**Conclusion**: Matrix III **is not** in reduced row echelon form due to the 9 in the first row violating the second condition.

---

### Final Answer:
- Matrix I **is** in reduced row echelon form.
- Matrices II and III **are not** in reduced row echelon form.

Would you like more details or further clarifications?

Here are 5 related questions to expand on this topic:
1. What are the specific steps to convert a matrix to reduced row echelon form?
2. How is Gaussian elimination different from Gauss-Jordan elimination?
3. What are the real-world applications of reduced row echelon form in solving systems of equations?
4. Can a matrix have more than one row-reduced form?
5. How does the number of leading 1s in a matrix relate to the rank of the matrix?

**Tip**: When solving systems of equations using matrices, reducing a matrix to row-reduced form makes it easier to identify the solutions.

Which of the following matrices are in row-reduced form? (Note: The dotted vertical line in each matrix should be a single vertical line.)

Learn how to determine if a matrix is in reduced row echelon form (RREF). This problem covers conditions such as leading 1s, non-zero entries in columns, and row order. Suitable for students studying Linear Algebra at the college or advanced high school level.

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