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the last option is E.none of these
tell me the correct option

The problem asks for expressing the matrix \( M \) as a product of elementary matrices based on the given row reduction steps. The original matrix \( M \) is:

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

The row operations performed are:
1. \( R_1 \leftrightarrow R_2 \) (Swap row 1 and row 2)
2. \( R_3 \rightarrow R_3 - R_1 \)
3. \( R_2 \rightarrow \frac{1}{2}R_2 \) (Multiply row 2 by \( \frac{1}{2} \))
4. \( R_3 \rightarrow R_3 - R_2 \)
5. \( R_1 \rightarrow R_1 - R_3 \)

Now, let's analyze how each of these operations affects the matrix \( M \) and its corresponding elementary matrices.

### 1. \( R_1 \leftrightarrow R_2 \)
Swapping rows 1 and 2 corresponds to multiplying \( M \) by the elementary matrix:

\[
E_1 = \begin{pmatrix} 
0 & 1 & 0 \\ 
1 & 0 & 0 \\ 
0 & 0 & 1 
\end{pmatrix}
\]

### 2. \( R_3 \rightarrow R_3 - R_1 \)
Subtracting row 1 from row 3 corresponds to multiplying \( M \) by the elementary matrix:

\[
E_2 = \begin{pmatrix} 
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
-1 & 0 & 1 
\end{pmatrix}
\]

### 3. \( R_2 \rightarrow \frac{1}{2}R_2 \)
Multiplying row 2 by \( \frac{1}{2} \) corresponds to multiplying \( M \) by the elementary matrix:

\[
E_3 = \begin{pmatrix} 
1 & 0 & 0 \\ 
0 & \frac{1}{2} & 0 \\ 
0 & 0 & 1 
\end{pmatrix}
\]

### 4. \( R_3 \rightarrow R_3 - R_2 \)
Subtracting row 2 from row 3 corresponds to multiplying \( M \) by the elementary matrix:

\[
E_4 = \begin{pmatrix} 
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & -1 & 1 
\end{pmatrix}
\]

### 5. \( R_1 \rightarrow R_1 - R_3 \)
Subtracting row 3 from row 1 corresponds to multiplying \( M \) by the elementary matrix:

\[
E_5 = \begin{pmatrix} 
1 & 0 & -1 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{pmatrix}
\]

### Final Expression:
The product of these elementary matrices gives the overall transformation, so the matrix \( M \) can be written as:

\[
A = E_5 E_4 E_3 E_2 E_1 M
\]

After analyzing the options provided (A, B, C, D, and E), none of the options seem to match this exact product. Therefore, **option E (None of these)** is the correct answer.

Let me know if you'd like further clarification on any of these steps or if you have additional questions!

Here are five follow-up questions:
1. How do elementary matrices relate to row operations?
2. Why do row swaps correspond to non-diagonal elementary matrices?
3. What is the determinant of an elementary matrix that multiplies a row by a scalar?
4. How do row reductions affect the rank of a matrix?
5. Can you express the inverse of a matrix using elementary matrices?

**Tip**: To find the inverse of a matrix, apply the same row operations used to reduce it to the identity matrix and track the changes in an identity matrix alongside.

Using these elementary row operations, how can we express the matrix M as a product of elementary matrices?

Explore how to express a matrix M as a product of elementary matrices through row reduction techniques. This problem demonstrates key matrix algebra concepts using elementary matrix products.

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