https://file.aiquickdraw.com/mathbot/file/1729247901024u46o73cf.jpg

The problem in your image involves matrix row operations. You are being asked to perform elementary row operations on a matrix.

The matrix is:

\[
\begin{bmatrix}
1 & 2 & 6 & -2 \\
3 & 1 & -1 & -2 \\
6 & 1 & -7 & 2
\end{bmatrix}
\]

The row operation provided is: \( -2R_2 + R_3 \rightarrow R_3 \), meaning you will multiply the second row by -2, and then add it to the third row to update the third row.

Let's work through the steps.

### Step 1: Multiply Row 2 by -2
\[
-2R_2 = -2 \times \begin{bmatrix} 3 & 1 & -1 & -2 \end{bmatrix} = \begin{bmatrix} -6 & -2 & 2 & 4 \end{bmatrix}
\]

### Step 2: Add this result to Row 3
\[
R_3 = \begin{bmatrix} 6 & 1 & -7 & 2 \end{bmatrix} + \begin{bmatrix} -6 & -2 & 2 & 4 \end{bmatrix} = \begin{bmatrix} 0 & -1 & -5 & 6 \end{bmatrix}
\]

### The updated matrix is:
\[
\begin{bmatrix}
1 & 2 & 6 & -2 \\
3 & 1 & -1 & -2 \\
0 & -1 & -5 & 6
\end{bmatrix}
\]

This is the matrix that should go in the blank after applying the indicated row operation.

Let me know if you'd like a detailed breakdown of each step, or if you have any further questions!

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### Questions:
1. What is the purpose of elementary row operations in matrix manipulation?
2. How does row reduction relate to solving systems of linear equations?
3. What are the different types of elementary row operations?
4. How do you know which row operation to use for Gaussian elimination?
5. How do you verify if a row operation was performed correctly?

### Tip:
When performing row operations, always keep the original matrix visible to avoid errors when updating specific rows.

Fill in the blank by performing the indicated elementary row operation(s): -2R2 + R3 -> R3 on the given matrix [ [1, 2, 6, -2], [3, 1, -1, -2], [6, 1, -7, 2] ]

Learn how to perform elementary row operations on a 3x4 matrix. This problem covers essential concepts in linear algebra such as row reduction and Gaussian elimination, suitable for undergraduate or advanced high school students.

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