Sure, let's go through the detailed steps to perform the row operation [tex]\( R_2 = -9R_1 + R_2 \)[/tex] on the given augmented matrix:
[tex]\[
\left[\begin{array}{rrr|r}
1 & 1 & -9 & -4 \\
9 & 6 & 8 & 1 \\
-5 & 6 & 7 & 4
\end{array}\right]
\][/tex]
### Step 1: Identify the Row Operation
We're instructed to perform the row operation on the second row [tex]\( R_2 \)[/tex]:
[tex]\[
R_2 = -9R_1 + R_2
\][/tex]
### Step 2: Multiply [tex]\( R_1 \)[/tex] by [tex]\(-9\)[/tex]
First, we need to multiply each element of the first row ([tex]\( R_1 \)[/tex]) by [tex]\(-9\)[/tex]:
[tex]\[
R_1 = \left[\begin{array}{rrrr}
1 & 1 & -9 & -4
\end{array}\right]
\][/tex]
[tex]\[
-9 \times R_1 = \left[\begin{array}{rrrr}
-9 \times 1 & -9 \times 1 & -9 \times (-9) & -9 \times (-4)
\end{array}\right] = \left[\begin{array}{rrrr}
-9 & -9 & 81 & 36
\end{array}\right]
\][/tex]
### Step 3: Add the Result to [tex]\( R_2 \)[/tex]
Now, we add this result to the elements of the second row [tex]\( R_2 \)[/tex]:
[tex]\[
R_2 = \left[\begin{array}{rrrr}
9 & 6 & 8 & 1
\end{array}\right]
\][/tex]
[tex]\[
R_2 = -9R_1 + R_2 = \left[\begin{array}{rrrr}
-9 & -9 & 81 & 36
\end{array}\right] + \left[\begin{array}{rrrr}
9 & 6 & 8 & 1
\end{array}\right]
\][/tex]
Combining the elements, we get:
[tex]\[
R_2 = \left[\begin{array}{rrrr}
-9 + 9 & -9 + 6 & 81 + 8 & 36 + 1
\end{array}\right] = \left[\begin{array}{rrrr}
0 & -3 & 89 & 37
\end{array}\right]
\][/tex]
### Step 4: Substitute Back into the Matrix
Finally, we substitute the new [tex]\( R_2 \)[/tex] back into the original matrix:
[tex]\[
\left[\begin{array}{rrr|r}
1 & 1 & -9 & -4 \\
0 & -3 & 89 & 37 \\
-5 & 6 & 7 & 4
\end{array}\right]
\][/tex]
Thus, after performing the row operation [tex]\( R_2 = -9R_1 + R_2 \)[/tex], the augmented matrix becomes:
[tex]\[
\left[\begin{array}{rrr|r}
1 & 1 & -9 & -4 \\
0 & -3 & 89 & 37 \\
-5 & 6 & 7 & 4
\end{array}\right]
\][/tex]
