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: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
• Apply the operation: 1.25 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

2. $R_3 + R_2 \rightarrow R_3$

• Start with the identity matrix: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
• Apply the operation: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}$$

3. $R_2 \rightarrow R_2 - R_3$

• Start with the identity matrix: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
• Apply the operation: 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix}$$

4. $0.8R_3 \rightarrow R_3$

• Start with the identity matrix: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
• Apply the operation: 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}$$

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.