Which of the following matrices is/are in reduced row-echelon form

$A=\begin{pmatrix}1&2&0\\0&1&0 \\0&0&1\end{pmatrix}$,

$B=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}$,

$C=\begin{pmatrix}1&0&0\\0&2&0\\0&0&1\end{pmatrix}$,

$D=\begin{pmatrix}1&2&0\\0&0&1\\0&0&0\\0&0&0\end{pmatrix}$?

To determine which matrices are in reduced row-echelon form (RREF), we need to verify that each matrix satisfies the following criteria:

1. **Leading entry**: The leading entry (first non-zero number from the left) in each non-zero row is 1.
2. **Column zeroes**: Each leading 1 is the only non-zero entry in its column.
3. **Row order**: The leading 1 in a row is to the right of the leading 1 in the row above it.
4. **Row of zeroes**: Any row consisting entirely of zeroes is at the bottom of the matrix.

Let's analyze each matrix:

### Matrix $ A $:
\[ A = \begin{pmatrix}1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} \]

1. Leading entries are 1.
2. Leading 1's are the only non-zero entries in their columns.
3. Each leading 1 is to the right of the leading 1 in the row above it.
4. No row of all zeroes.

**Matrix $ A $ is in RREF.**

### Matrix $ B $:
\[ B = \begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix} \]

1. The first row has a leading 1, but it's not in the first column.
2. Leading 1's are the only non-zero entries in their columns.
3. The leading 1 in the second row is not to the right of the leading 1 in the first row.

**Matrix $ B $ is not in RREF.**

### Matrix $ C $:
\[ C = \begin{pmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\end{pmatrix} \]

1. The first row has a leading 1.
2. The second row has a leading entry of 2 (not 1), which violates the first condition.
3. The third row is fine but the violation in the second row is enough.

**Matrix $ C $ is not in RREF.**

### Matrix $ D $:
\[ D = \begin{pmatrix}1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} \]

1. Leading entries are 1.
2. Leading 1's are the only non-zero entries in their columns.
3. Each leading 1 is to the right of the leading 1 in the row above it.
4. Rows of all zeroes are at the bottom.

**Matrix $ D $ is in RREF.**

### Conclusion
The matrices in reduced row-echelon form are:

\[ A = \begin{pmatrix}1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} \]
\[ D = \begin{pmatrix}1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} \]

Would you like further details or have any questions?

Here are 5 questions you might ask next:
1. How do you perform Gaussian elimination to achieve RREF?
2. What is the difference between row-echelon form and reduced row-echelon form?
3. Can you provide an example of a matrix not in RREF and show how to convert it?
4. What are the applications of reduced row-echelon form in solving linear systems?
5. How do you determine the rank of a matrix from its RREF?

**Tip:** In reduced row-echelon form, each leading 1 must be the only non-zero entry in its column, ensuring that each row and column maintains a specific order.

Learn how to identify matrices in reduced row-echelon form (RREF) with detailed explanations and examples. This problem covers key concepts in linear algebra, including row operations and the criteria for RREF. Suitable for advanced high school students and college-level learners.

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