To solve this problem and find out which matrix results from the operation [tex]\( -3 R_2 \leftrightarrow R_2 \)[/tex], let's break down the steps one by one.
We start with the given augmented matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 1 & -1 \\
1 & 3 & -1 & -9 \\
3 & 2 & 0 & -2
\end{array}\right]
\][/tex]
The operation [tex]\( -3 R_2 \leftrightarrow R_2 \)[/tex] instructs us to multiply the entire second row by [tex]\(-3\)[/tex].
Let's apply this operation to the second row:
- The first element of the second row is [tex]\(1\)[/tex]. After multiplying by [tex]\(-3\)[/tex], we get [tex]\(-3 \cdot 1 = -3\)[/tex].
- The second element of the second row is [tex]\(3\)[/tex]. After multiplying by [tex]\(-3\)[/tex], we get [tex]\(-3 \cdot 3 = -9\)[/tex].
- The third element of the second row is [tex]\(-1\)[/tex]. After multiplying by [tex]\(-3\)[/tex], we get [tex]\(-3 \cdot (-1) = 3\)[/tex].
- The fourth element of the second row (the augmented part) is [tex]\(-9\)[/tex]. After multiplying by [tex]\(-3\)[/tex], we get [tex]\(-3 \cdot (-9) = 27\)[/tex].
Hence, the second row after the operation becomes [tex]\([-3, -9, 3, 27]\)[/tex].
Now, we construct the new augmented matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 1 & -1 \\
-3 & -9 & 3 & 27 \\
3 & 2 & 0 & -2
\end{array}\right]
\][/tex]
Comparing this with the given options:
1. [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 1 & -1 \\ -3 & -9 & 3 & 27 \\ 3 & 2 & 0 & -2\end{array}\right]\)[/tex]
2. [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 1 & -1 \\ -3 & -3 & -3 & -3 \\ 3 & 2 & 0 & -2\end{array}\right]\)[/tex]
We see that the resulting matrix is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 1 & -1 \\
-3 & -9 & 3 & 27 \\
3 & 2 & 0 & -2
\end{array}\right]
\][/tex]
Therefore, the correct resulting matrix from the operation [tex]\( -3 R_2 \leftrightarrow R_2 \)[/tex] is:
[tex]\(\left[\begin{array}{ccc|c}1 & 0 & 1 & -1 \\ -3 & -9 & 3 & 27 \\ 3 & 2 & 0 & -2\end{array}\right]\)[/tex]
