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The image shows a matrix operation problem where you are required to perform elementary row operations on the given matrix. The operations indicated are:

- \( R_3 + R_1 \) (add row 3 to row 1)
- \( -3R_3 + R_2 \) (multiply row 3 by -3 and then add it to row 2)

Let's first write down the given matrix and apply the indicated operations step by step:

The given matrix is:

\[
\begin{pmatrix}
3 & -9 & 3 & -6 \\
1 & -3 & 1 & -2 \\
-5 & -8 & 4 & 7
\end{pmatrix}
\]

### Step 1: Apply the first operation \( R_3 + R_1 \) (add row 3 to row 1)
Performing \( R_1 = R_3 + R_1 \):

\[
R_1 = (3, -9, 3, -6) + (-5, -8, 4, 7) = (-2, -17, 7, 1)
\]

The new matrix is:

\[
\begin{pmatrix}
-2 & -17 & 7 & 1 \\
1 & -3 & 1 & -2 \\
-5 & -8 & 4 & 7
\end{pmatrix}
\]

### Step 2: Apply the second operation \( -3R_3 + R_2 \) (multiply row 3 by -3, then add it to row 2)
First, multiply row 3 by -3:

\[
-3R_3 = (-3)(-5, -8, 4, 7) = (15, 24, -12, -21)
\]

Now, add this result to row 2:

\[
R_2 = (1, -3, 1, -2) + (15, 24, -12, -21) = (16, 21, -11, -23)
\]

The final matrix is:

\[
\begin{pmatrix}
-2 & -17 & 7 & 1 \\
16 & 21 & -11 & -23 \\
-5 & -8 & 4 & 7
\end{pmatrix}
\]

So, the resulting matrix after performing the indicated row operations is:

\[
\boxed{
\begin{pmatrix}
-2 & -17 & 7 & 1 \\
16 & 21 & -11 & -23 \\
-5 & -8 & 4 & 7
\end{pmatrix}
}
\]

Let me know if you need further details, or if you have any questions!

Here are some related questions:
1. How can Gaussian elimination be used to solve systems of linear equations?
2. What are the different types of elementary row operations?
3. How do elementary row operations affect the determinant of a matrix?
4. Can elementary row operations change the rank of a matrix?
5. What is the purpose of reducing a matrix to row echelon form?

**Tip:** Always double-check the arithmetic when performing row operations, as small mistakes can lead to incorrect solutions.

Perform elementary row operations on the given matrix: [[3, -9, 3, -6], [1, -3, 1, -2], [-5, -8, 4, 7]]. The operations are: -3R_3 + R_1 and -3R_3 + R_2.

Solve matrix row operations using Gaussian elimination. This problem covers elementary row operations applied to a 3x4 matrix. Learn step-by-step how to modify matrix rows and solve the system. Suitable for high school or college-level students.

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