https://file.aiquickdraw.com/mathbot/file/17297846089348lk7eunb.jpg

Let's solve the problem step by step. 

### Part a)
We are starting with the following augmented matrix:

\[
\begin{bmatrix}
1 & 3 & 0 & | & 1 & 0 & 0 \\
-4 & -1 & 2 & | & 0 & 1 & 0 \\
1 & 7 & -5 & | & 0 & 0 & 1 \\
\end{bmatrix}
\]

We are instructed to perform the row operation:

\[
R_2 = R_2 + 4R_1
\]

This means we add 4 times the first row to the second row. Let's compute the result for the second row:

\[
R_2 = (-4, -1, 2 | 0, 1, 0) + 4 \cdot (1, 3, 0 | 1, 0, 0)
\]

Performing the calculations:

- Second row: 
  \[
  (-4 + 4 \cdot 1, -1 + 4 \cdot 3, 2 + 4 \cdot 0 | 0 + 4 \cdot 1, 1 + 4 \cdot 0, 0 + 4 \cdot 0)
  \]

  Which gives:

  \[
  (0, 11, 2 | 4, 1, 0)
  \]

Now the augmented matrix looks like this:

\[
\begin{bmatrix}
1 & 3 & 0 & | & 1 & 0 & 0 \\
0 & 11 & 2 & | & 4 & 1 & 0 \\
1 & 7 & -5 & | & 0 & 0 & 1 \\
\end{bmatrix}
\]

Here, \( a = 11 \), \( b = 2 \), \( c = 4 \), and \( d = 1 \).

### Part b)
Next, we perform the operation:

\[
R_3 = R_3 - R_1
\]

This means we subtract the first row from the third row. Let's compute the result for the third row:

\[
R_3 = (1, 7, -5 | 0, 0, 1) - (1, 3, 0 | 1, 0, 0)
\]

Performing the calculations:

- Third row: 
  \[
  (1 - 1, 7 - 3, -5 - 0 | 0 - 1, 0 - 0, 1 - 0)
  \]

  Which gives:

  \[
  (0, 4, -5 | -1, 0, 1)
  \]

Now the augmented matrix looks like this:

\[
\begin{bmatrix}
1 & 3 & 0 & | & 1 & 0 & 0 \\
0 & 11 & 2 & | & 4 & 1 & 0 \\
0 & 4 & -5 & | & -1 & 0 & 1 \\
\end{bmatrix}
\]

Here, \( e = 4 \), \( f = -5 \), and \( g = -1 \).

---

Let me know if you would like more details or have any questions. Here are 5 related questions to further explore the topic:

1. How do you continue the process to find the inverse of this matrix?
2. What is the significance of performing row operations in finding the inverse of a matrix?
3. Can every matrix have an inverse? If not, what conditions must hold for a matrix to be invertible?
4. What would happen if a matrix had a row of zeros during this process?
5. How can you verify whether your result for the inverse is correct?

**Tip:** When performing row operations, always double-check each arithmetic step to avoid small errors that could affect the final result.

Perform the row operations R2 = R2 + 4R1 and R3 = R3 - R1 on the given augmented matrix and identify the values of a, b, c, d, e, f, and g.

Learn how to perform row operations on an augmented matrix to find the inverse of a 3x3 matrix. This step-by-step guide covers key linear algebra concepts such as matrix inversion, elementary row operations, and solving systems of equations.

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