Let's perform the indicated row operations on the given augmented matrix.
The original matrix is:
[tex]\[
\left[\begin{array}{ccc|c}
2 & -1 & -4 & 7 \\
0 & 3 & -2 & 5 \\
0 & 10 & 8 & 2
\end{array}\right]
\][/tex]
The row operation to be performed is: [tex]\( -10 \cdot R2 \rightarrow R2 \)[/tex].
### Step-by-Step Solution
Step 1: Label the Rows
- [tex]\( R1 = [2, -1, -4, 7] \)[/tex]
- [tex]\( R2 = [0, 3, -2, 5] \)[/tex]
- [tex]\( R3 = [0, 10, 8, 2] \)[/tex]
Step 2: Perform the Row Operation
We need to multiply the second row by [tex]\(-10\)[/tex] and then add the result to the third row. This can be broken into the following sub-steps:
- Multiply the second row ([tex]\( R2 \)[/tex]) by [tex]\(-10\)[/tex]:
[tex]\[
-10 \cdot R2 = -10 \cdot [0, 3, -2, 5] = [0, -30, 20, -50]
\][/tex]
- Add this result to the third row ([tex]\( R3 \)[/tex]):
[tex]\[
R3 + (-10 \cdot R2) = [0, 10, 8, 2] + [0, -30, 20, -50]
\][/tex]
Perform the element-wise addition:
[tex]\[
[0 + 0, 10 + (-30), 8 + 20, 2 + (-50)] = [0, -20, 28, -48]
\][/tex]
So, the modified third row [tex]\( R3 \)[/tex] becomes [tex]\([0, -20, 28, -48]\)[/tex].
Step 3: Update the Matrix
The updated matrix now looks like this:
[tex]\[
\left[\begin{array}{ccc|c}
2 & -1 & -4 & 7 \\
0 & 3 & -2 & 5 \\
0 & -20 & 28 & -48
\end{array}\right]
\][/tex]
This is the final result after performing the indicated row operation.
