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The problem you provided involves a Gauss-Jordan reduction process to find elementary matrices that transform matrix \( A \) into the identity matrix \( I \). The elementary matrices \( E_1, E_2, E_3, E_4 \) are derived from the steps performed in the reduction process.

Let's break down the transformation steps shown in the image:

1. **Initial matrix \( A \):**
   \[
   A = \begin{bmatrix} 
   0 & 1 & 6 \\
   -9 & -27 & 0 \\
   0 & 0 & 1 
   \end{bmatrix}
   \]

2. **Step 1 (Operation resulting in \( E_1 A \)):**
   The first step seems to perform a row operation that changes the second row. Likely, \( R_2 \to R_2 + 9R_1 \), which adds 9 times the first row to the second row, yielding:
   \[
   E_1 A = \begin{bmatrix} 
   0 & 1 & 6 \\
   0 & -9 & 54 \\
   0 & 0 & 1 
   \end{bmatrix}
   \]
   The corresponding elementary matrix \( E_1 \) performs this row operation.

3. **Step 2 (Operation resulting in \( E_2 E_1 A \)):**
   Here, the second row seems to be scaled. \( R_2 \to \frac{1}{-9} R_2 \), giving us:
   \[
   E_2 E_1 A = \begin{bmatrix} 
   0 & 1 & 6 \\
   0 & 1 & -6 \\
   0 & 0 & 1 
   \end{bmatrix}
   \]
   The corresponding elementary matrix \( E_2 \) scales the second row.

4. **Step 3 (Operation resulting in \( E_3 E_2 E_1 A \)):**
   Next, we perform an operation to eliminate the second row's effect from the first row. Likely, \( R_1 \to R_1 - R_2 \), yielding:
   \[
   E_3 E_2 E_1 A = \begin{bmatrix} 
   0 & 1 & 0 \\
   0 & 1 & -6 \\
   0 & 0 & 1 
   \end{bmatrix}
   \]
   This eliminates the \( 6 \) in the first row, third column.

5. **Step 4 (Operation resulting in \( E_4 E_3 E_2 E_1 A \)):**
   Finally, we need to swap the first two rows to ensure the identity matrix structure. This row exchange results in:
   \[
   E_4 E_3 E_2 E_1 A = \begin{bmatrix} 
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1 
   \end{bmatrix}
   \]
   This is the identity matrix \( I \).

### Finding the elementary matrices \( E_1, E_2, E_3, E_4 \):
Each elementary matrix corresponds to the row operation applied in the process. To find them:

- **\( E_1 \)** corresponds to adding 9 times row 1 to row 2.
- **\( E_2 \)** scales row 2 by \( -\frac{1}{9} \).
- **\( E_3 \)** subtracts row 2 from row 1.
- **\( E_4 \)** swaps rows 1 and 2.

Now, you can multiply the inverses of these matrices to obtain \( A \).

Would you like me to compute the exact forms of \( E_1, E_2, E_3, E_4 \) and their inverses?

Find the elementary matrices E1, E2, E3, E4 involved in the Gauss-Jordan reduction of matrix A to the identity matrix, and express matrix A as the product of these elementary matrices and their inverses.

Explore the Gauss-Jordan reduction process to find elementary matrices that transform matrix A into the identity matrix. Learn how to express matrix A as the product of these elementary matrices and their inverses, with a step-by-step breakdown of the operations involved.

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