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a. Find elementary matrices E1, E2, E3, and E4 that perform these row operations.

To solve this problem, we need to determine the elementary matrices \( E_1, E_2, E_3, \) and \( E_4 \) corresponding to the given row operations. 

### Understanding the Problem

Given a sequence of row operations, elementary matrices represent these operations when multiplied by the original matrix. Let's break down the operations applied to the identity matrix \( I \) to find each elementary matrix.

The row operations shown in the image are as follows:

1. \( 1.25R_1 \rightarrow R_1 \)  
   This operation scales the first row by 1.25.

2. \( R_3 + R_2 \rightarrow R_3 \)  
   This operation adds the second row to the third row.

3. \( R_2 \rightarrow R_2 - R_3 \)  
   This operation subtracts the third row from the second row.

4. \( 0.8R_3 \rightarrow R_3 \)  
   This operation scales the third row by 0.8.

### Finding the Elementary Matrices

Each elementary matrix is obtained by applying the corresponding row operation to the identity matrix.

#### 1. \( 1.25R_1 \rightarrow R_1 \)
- Start with the identity matrix:
  \[
  I = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]
- Apply the operation:
  \[
  E_1 = \begin{pmatrix} 
  1.25 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]

#### 2. \( R_3 + R_2 \rightarrow R_3 \)
- Start with the identity matrix:
  \[
  I = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]
- Apply the operation:
  \[
  E_2 = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 1 & 1 
  \end{pmatrix}
  \]

#### 3. \( R_2 \rightarrow R_2 - R_3 \)
- Start with the identity matrix:
  \[
  I = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]
- Apply the operation:
  \[
  E_3 = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & -1 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]

#### 4. \( 0.8R_3 \rightarrow R_3 \)
- Start with the identity matrix:
  \[
  I = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 1 
  \end{pmatrix}
  \]
- Apply the operation:
  \[
  E_4 = \begin{pmatrix} 
  1 & 0 & 0 \\ 
  0 & 1 & 0 \\ 
  0 & 0 & 0.8 
  \end{pmatrix}
  \]

### Summary

The elementary matrices \( E_1, E_2, E_3, \) and \( E_4 \) are:
- \( E_1 = \begin{pmatrix} 1.25 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
- \( E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix} \)
- \( E_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix} \)
- \( E_4 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0.8 \end{pmatrix} \)

Would you like a detailed explanation on any specific part of this? Here are some questions to consider:

1. What is the role of an elementary matrix in linear algebra?
2. How can you determine the inverse of an elementary matrix?
3. How does multiplying a matrix by an elementary matrix affect its determinant?
4. Can a sequence of elementary matrices represent any invertible matrix?
5. How would the process differ if one of the row operations was swapping two rows?

**Tip:** Elementary matrices are extremely useful for performing Gaussian elimination and finding the inverse of matrices efficiently.

Learn how to find elementary matrices E1, E2, E3, and E4 that correspond to given row operations. This problem explores linear algebra concepts related to elementary matrices and their application in transforming matrices through row operations. Suitable for undergraduate students studying linear algebra.

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