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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?

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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.

How can we express matrix A as a product of elementary matrices, given the row reduction of matrix M?

Learn how to express matrix A as a product of elementary matrices through a step-by-step solution of matrix row reduction operations on matrix M. The problem involves key linear algebra concepts like elementary matrices and row operations. Suitable for university-level linear algebra courses.

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