https://file.aiquickdraw.com/mathbot/file/1728350147192wgrw5731.jpg

The image shows a matrix row reduction process for matrix \( M = \begin{pmatrix} 0 & 2 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{pmatrix} \), followed by several steps of elementary row operations. The goal is to express matrix \( A \) (the reduced matrix after row operations) as a product of elementary matrices, and we are given four possible expressions for \( M \) as a product of elementary matrices.

### Step-by-step Row Operations:

1. **First operation**: \( R_1 \leftrightarrow R_2 \) (swap row 1 and row 2)
   \[
   \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 1 & 2 \end{pmatrix}
   \]

2. **Second operation**: \( R_3 \to R_3 - R_1 \) (subtract row 1 from row 3)
   \[
   \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 1 \end{pmatrix}
   \]

3. **Third operation**: \( R_2 \to \frac{1}{2} R_2 \) (scale row 2 by \( \frac{1}{2} \))
   \[
   \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}
   \]

4. **Fourth operation**: \( R_3 \to R_3 - R_2 \) (subtract row 2 from row 3)
   \[
   \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
   \]

5. **Fifth operation**: \( R_1 \to R_1 - R_3 \) (subtract row 3 from row 1)
   \[
   \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} 
   = I_3
   \]
   
After these row operations, the matrix is reduced to the identity matrix, which means that \( M \) can be expressed as the product of elementary matrices corresponding to these operations.

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

The elementary matrices corresponding to the operations can be written down as follows:
- First operation \( R_1 \leftrightarrow R_2 \): Swap rows 1 and 2.
- Second operation \( R_3 - R_1 \): Subtract row 1 from row 3.
- Third operation \( \frac{1}{2} R_2 \): Scale row 2 by \( \frac{1}{2} \).
- Fourth operation \( R_3 - R_2 \): Subtract row 2 from row 3.
- Fifth operation \( R_1 - R_3 \): Subtract row 3 from row 1.

Looking at the options given (A, B, C, and D), we can now evaluate which one matches the series of operations.

### The correct answer is **A**.

Would you like to dive deeper into how the elementary matrices are constructed for each step, or do you have any other questions?

Here are some related questions:
1. What are the general properties of elementary matrices?
2. How can we find the inverse of an elementary matrix?
3. Why does row reduction lead to the identity matrix for invertible matrices?
4. What is the significance of row operations in solving linear systems?
5. How does LU decomposition relate to elementary row operations?

**Tip**: When performing matrix row operations, the same operations must be applied consistently to the identity matrix to form the corresponding elementary matrices.

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

Explore how matrix M is reduced using row operations and express the final matrix as a product of elementary matrices. This detailed solution covers concepts such as row swaps, row scaling, and matrix inversion, essential for students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation