Let's go through the problem step by step:

We are given a matrix $M$ and a sequence of elementary row operations that transform $M$ into the identity matrix. Our task is to express the matrix $A$, which is the identity matrix, as a product of elementary matrices that correspond to those operations.

The given row reduction operations:

The original matrix $M$ is: $M = \begin{pmatrix} 0 & 2 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{pmatrix}$

The row operations are:

1. $R_1 \leftrightarrow R_2$: Swap row 1 with row 2. $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 1 & 2 \end{pmatrix}$

2. $R_3 \rightarrow R_3 - R_1$: Subtract row 1 from row 3. $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 1 \end{pmatrix}$

3. $R_2 \rightarrow \frac{1}{2} R_2$: Divide row 2 by 2. $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}$

4. $R_3 \rightarrow R_3 - R_2$: Subtract row 2 from row 3. $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

5. $R_1 \rightarrow R_1 - R_3$: Subtract row 3 from row 1. $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

After these operations, we get the identity matrix $I_3$.

Step-by-step analysis of each operation:

Each of these operations corresponds to an elementary matrix. To find the product of elementary matrices, we create a matrix for each of the row operations:

1. $R_1 \leftrightarrow R_2$: This is a row swap matrix, so the elementary matrix is: $E_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

2. $R_3 \rightarrow R_3 - R_1$: This corresponds to subtracting row 1 from row 3: $E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}$

3. $R_2 \rightarrow \frac{1}{2} R_2$: This is scaling row 2 by $\frac{1}{2}$: $E_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}$

4. $R_3 \rightarrow R_3 - R_2$: This corresponds to subtracting row 2 from row 3: $E_4 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$

5. $R_1 \rightarrow R_1 - R_3$: This corresponds to subtracting row 3 from row 1: $E_5 = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Now, multiplying these elementary matrices together:

$A = E_5 E_4 E_3 E_2 E_1$

Conclusion:

By comparing the given options with the product of elementary matrices, Option A is correct. It correctly expresses the matrix $A$ as a product of the elementary matrices corresponding to the row operations performed.

Would you like more details about how to multiply these matrices, or do you have any other questions?

Here are 5 related questions for further practice:

1. How do elementary row operations affect the determinant of a matrix?
2. What is the significance of an elementary matrix in matrix theory?
3. How can row reduction be used to compute the inverse of a matrix?
4. How do you identify the elementary matrix corresponding to a specific row operation?
5. Can every square matrix be reduced to the identity matrix using elementary row operations?

Tip: To find the inverse of a matrix, use elementary row operations to transform the matrix into the identity matrix, while applying the same operations to the identity matrix.