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The image shows a matrix operation where an elementary row operation is being performed on a 2x3 augmented matrix. The operation indicated is `-4R2 + R1`.

Let's break down the operation:

- \( R_1 \) is the first row of the matrix.
- \( R_2 \) is the second row of the matrix.
- The operation is \( R_1 \) → \( -4R_2 + R_1 \), which means we multiply the entire second row by -4 and then add it to the first row.

The given matrix is:

\[
\begin{pmatrix}
4 & -7 & 6 \\
1 & 6 & -1
\end{pmatrix}
\]

Now, let's perform the operation step-by-step:

1. Multiply the second row \( R_2 \) by -4:
   \[
   -4R_2 = -4 \times \begin{pmatrix} 1 & 6 & -1 \end{pmatrix} = \begin{pmatrix} -4 & -24 & 4 \end{pmatrix}
   \]

2. Add this result to the first row \( R_1 \):
   \[
   R_1 + (-4R_2) = \begin{pmatrix} 4 & -7 & 6 \end{pmatrix} + \begin{pmatrix} -4 & -24 & 4 \end{pmatrix} = \begin{pmatrix} 0 & -31 & 10 \end{pmatrix}
   \]

So, after performing the operation \( R_1 \rightarrow -4R_2 + R_1 \), the new matrix is:

\[
\begin{pmatrix}
0 & -31 & 10 \\
1 & 6 & -1
\end{pmatrix}
\]

This is the resulting matrix after applying the indicated elementary row operation.

Do you want any additional details or have any questions?

Here are some related questions to deepen your understanding:

1. What are the types of elementary row operations and how do they affect a matrix?
2. How can elementary row operations be used to find the inverse of a matrix?
3. What does the determinant of a matrix signify, and how is it affected by row operations?
4. Can you explain how row reduction can help solve systems of linear equations?
5. How can Gaussian elimination be used to bring a matrix to its reduced row echelon form?

**Tip:** When performing elementary row operations, it's crucial to follow the steps carefully to avoid errors, especially when dealing with larger matrices.

Perform the indicated elementary row operation on the given augmented matrix: -4R2 + R1.

Learn how to apply the elementary row operation -4R2 + R1 to a 2x3 augmented matrix. This step-by-step explanation breaks down the process of Gaussian elimination and demonstrates how row reduction can help solve systems of linear equations. Suitable for students in Grades 10-12.

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