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.