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In this image, the matrix \( A \) is transformed into \( E_1 A \) through the application of an elementary row operation as part of the Gaussian-Jordan method.

We need to find the elementary matrix \( E_1 \), which performs this row operation. Let's analyze the transition from matrix \( A \) to \( E_1 A \).

The matrix \( A \) is:
\[
A = \begin{bmatrix}
1 & -6 & 0 \\
-21 & 0 & 3 \\
0 & 1 & 0
\end{bmatrix}
\]

The resulting matrix \( E_1 A \) is:
\[
E_1 A = \begin{bmatrix}
1 & 0 & 0 \\
-21 & 0 & 3 \\
0 & 1 & 0
\end{bmatrix}
\]

### Step-by-step process using Gaussian-Jordan method:

1. **Look at the first row transformation**: We see that row 1 in matrix \( A \) has changed from \([1, -6, 0]\) to \([1, 0, 0]\) in matrix \( E_1 A \). This means that a row operation was applied to set the second element of the first row to 0.

2. **Determine the elementary row operation**: To transform the first row of matrix \( A \), the row operation must have been to add 6 times the third row to the first row (i.e., \( R_1 \rightarrow R_1 + 6R_3 \)).

Thus, \( E_1 \), the elementary matrix corresponding to this row operation, would be the identity matrix with the appropriate transformation applied to the first row:
\[
E_1 = \begin{bmatrix}
1 & 6 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

### Verification:
Multiplying \( E_1 \) by \( A \):

\[
E_1 A = \begin{bmatrix}
1 & 6 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & -6 & 0 \\
-21 & 0 & 3 \\
0 & 1 & 0
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 \\
-21 & 0 & 3 \\
0 & 1 & 0
\end{bmatrix}
\]

This confirms that \( E_1 \) is correct.

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:
1. What are the key differences between Gaussian elimination and Gaussian-Jordan method?
2. How does the form of an elementary matrix change based on the row operation applied?
3. Why is Gaussian-Jordan elimination used for finding the inverse of a matrix?
4. What happens if we apply row swaps instead of row additions in elementary matrices?
5. Can \( E_1 \) be applied to any matrix or is it specific to matrix \( A \)?

**Tip:** When performing Gaussian elimination, always track row operations carefully, as they directly determine the corresponding elementary matrix.

Find E_1 using the Gaussian-Jordan method.

Learn how to find the elementary matrix E_1 using the Gaussian-Jordan method. This problem demonstrates row operations on a matrix and the creation of an elementary matrix. Suitable for students studying linear algebra.

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