Which of the following matrices are in reduced row echelon form?

Select TWO correct answers.
A.
	1	0	6	
0	–1	1

B.
	1	0	0	2	
0	1	0	–5.5
0	0	1	9

C.
	1	0	6	
0	1	–1

D.
	1	0	0	2	
0	2	0	4
0	0	3	6

E.
	1	0	0	5	
0	1	1	3
0	0	1	6


To identify which matrices are in reduced row echelon form (RREF), we need to recall the following key properties of a matrix in RREF:

1. **Leading ones**: Each leading entry in a nonzero row must be 1 (this is called a pivot).
2. **Zeros above and below pivots**: The column containing a pivot (leading 1) must have zeros everywhere else (above and below the pivot).
3. **Row ordering**: Any row that contains only zeros is at the bottom of the matrix.
4. **Left-to-right progression**: The leading 1 of any row must appear to the right of the leading 1 in the row above it.

Now, let's analyze each matrix:

### Option A:
\[
\begin{bmatrix}
1 & 0 & 6 \\
0 & -1 & 1
\end{bmatrix}
\]
- The first row has a leading 1 in the first column. The second row has a leading entry of -1, but **this is not a 1**, so this matrix is **not** in RREF.

### Option B:
\[
\begin{bmatrix}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & -5.5 \\
0 & 0 & 1 & 9
\end{bmatrix}
\]
- The leading 1s are in positions (1,1), (2,2), and (3,3), and each of these leading 1s has zeros above and below them.
- All non-zero rows have their pivots to the right of those in the row above.
- This matrix **is** in RREF.

### Option C:
\[
\begin{bmatrix}
1 & 0 & 6 \\
0 & 1 & -1
\end{bmatrix}
\]
- The first row has a leading 1 in the first column, and the second row has a leading 1 in the second column.
- There are zeros above and below each leading 1.
- This matrix **is** in RREF.

### Option D:
\[
\begin{bmatrix}
1 & 0 & 0 & 2 \\
0 & 2 & 0 & 4 \\
0 & 0 & 3 & 6
\end{bmatrix}
\]
- The second row has a leading entry of 2 (not 1), and the third row has a leading entry of 3 (not 1).
- This matrix is **not** in RREF because of the non-1 leading entries.

### Option E:
\[
\begin{bmatrix}
1 & 0 & 0 & 5 \\
0 & 1 & 1 & 3 \\
0 & 0 & 1 & 6
\end{bmatrix}
\]
- The leading 1s are in positions (1,1), (2,2), and (3,3), and the matrix has zeros above and below these leading 1s.
- This matrix **is** in RREF.

### Conclusion:
The matrices in reduced row echelon form are:
- **Option B**.
- **Option E**.

Would you like more details or have any questions?

### 5 Relative Questions:
1. How does row echelon form differ from reduced row echelon form?
2. What are some efficient methods to compute the RREF of a matrix?
3. How does RREF relate to the solution of a system of linear equations?
4. Can a matrix with non-integer entries be in RREF?
5. How do elementary row operations affect a matrix?

### Tip:
When performing row reduction, always aim to create leading 1s and clear out all nonzero entries above and below these leading 1s as the first step toward RREF.

Which of the following matrices are in reduced row echelon form?
Select TWO correct answers.
A.
1 0 6
0 –1 1

B.
1 0 0 2
0 1 0 -5.5
0 0 1 9

C.
1 0 6
0 1 -1

D.
1 0 0 2
0 2 0 4
0 0 3 6

E.
1 0 0 5
0 1 1 3
0 0 1 6

Learn how to determine if a matrix is in reduced row echelon form (RREF). This problem focuses on key linear algebra concepts such as leading ones, zeros above and below pivots, and the structure of RREF matrices. Suitable for students in Grades 10-12.

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