https://file.aiquickdraw.com/mathbot/file/1728268390484nxqm4b4d.jpg

The image shows a matrix \( M \) and a series of row operations reducing it to the identity matrix. The problem asks how to express \( M \) as a product of elementary matrices based on the given row operations.

### Matrix \( M \):
\[
M = \begin{pmatrix} 
0 & 2 & 0 \\
1 & 0 & 1 \\
1 & 1 & 2 
\end{pmatrix}
\]

### The row operations:
1. \( R_1 \leftrightarrow R_2 \) (swap row 1 and row 2)
2. \( R_3 \rightarrow R_3 - R_1 \) (subtract row 1 from row 3)
3. \( \frac{1}{2} R_2 \) (scale row 2 by \( \frac{1}{2} \))
4. \( R_3 \rightarrow R_3 - R_2 \) (subtract row 2 from row 3)
5. \( R_1 \rightarrow R_1 - R_3 \) (subtract row 3 from row 1)

### To express \( M \) as a product of elementary matrices:

For each row operation, there is a corresponding elementary matrix. Let's derive the elementary matrices for each operation.

1. **Elementary matrix for \( R_1 \leftrightarrow R_2 \)**:
   The operation swaps the first and second rows. The corresponding elementary matrix \( E_1 \) is:
   \[
   E_1 = \begin{pmatrix}
   0 & 1 & 0 \\
   1 & 0 & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

2. **Elementary matrix for \( R_3 \rightarrow R_3 - R_1 \)**:
   The operation subtracts row 1 from row 3. The corresponding elementary matrix \( E_2 \) is:
   \[
   E_2 = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   -1 & 0 & 1
   \end{pmatrix}
   \]

3. **Elementary matrix for \( \frac{1}{2} R_2 \)**:
   The operation scales row 2 by \( \frac{1}{2} \). The corresponding elementary matrix \( E_3 \) is:
   \[
   E_3 = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & \frac{1}{2} & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

4. **Elementary matrix for \( R_3 \rightarrow R_3 - R_2 \)**:
   The operation subtracts row 2 from row 3. The corresponding elementary matrix \( E_4 \) is:
   \[
   E_4 = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & -1 & 1
   \end{pmatrix}
   \]

5. **Elementary matrix for \( R_1 \rightarrow R_1 - R_3 \)**:
   The operation subtracts row 3 from row 1. The corresponding elementary matrix \( E_5 \) is:
   \[
   E_5 = \begin{pmatrix}
   1 & 0 & -1 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

### Matrix \( M \) as a product of elementary matrices:
The matrix \( M \) can be expressed as the product of the elementary matrices as follows:
\[
M = E_5^{-1} E_4^{-1} E_3^{-1} E_2^{-1} E_1^{-1} I
\]
where \( I \) is the identity matrix, and \( E_1, E_2, E_3, E_4, E_5 \) are the elementary matrices corresponding to the row operations.

The inverses of the elementary matrices are simple to compute, as:
- The inverse of a row swap is the same row swap.
- The inverse of a scaling by \( k \) is scaling by \( \frac{1}{k} \).
- The inverse of a row operation involving subtraction is adding the same multiple back.

Thus:
\[
M = E_1 E_2 E_3 E_4 E_5
\]

Let me know if you would like further clarifications or details! 

Here are 5 questions to expand on this:

1. What are elementary matrices, and why are they important in matrix theory?
2. How do we verify the inverses of elementary matrices for each row operation?
3. What properties of the elementary matrices make them useful for expressing matrix factorizations?
4. Could we apply these operations to different types of matrices, such as non-square matrices?
5. How does this process connect to the LU decomposition of a matrix?

**Tip**: Elementary matrices are useful in solving systems of linear equations, performing matrix inversions, and simplifying matrices to reduced forms like row echelon form.

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

Learn how to express a matrix as a product of elementary matrices using a series of row operations. This problem covers key concepts in matrix theory and matrix factorization, including elementary row operations like swaps, row additions, and scaling.

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