To determine which matrix results from the given operation [tex]\(-3 R_2 \leftrightarrow R_2\)[/tex] on the original augmented matrix, we need to carefully apply the operation to the second row of the initial matrix.
Here is the original 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] indicates that each element in the second row should be multiplied by [tex]\(-3\)[/tex].
Let's perform this operation on the second row step by step:
- For the first element of the second row: [tex]\(1 \times -3 = -3\)[/tex]
- For the second element of the second row: [tex]\(3 \times -3 = -9\)[/tex]
- For the third element of the second row: [tex]\(-1 \times -3 = 3\)[/tex]
- For the fourth element of the second row (the augmented part): [tex]\(-9 \times -3 = 27\)[/tex]
So the new second row after applying [tex]\(-3 R_2 \leftrightarrow R_2\)[/tex] will be:
[tex]\[
[-3, -9, 3, 27]
\][/tex]
Thus, the resulting matrix will be:
[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 matrix resulting 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]
This corresponds to the first given option:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 1 & -1 \\
-3 & -9 & 3 & 27 \\
3 & 2 & 0 & -2
\end{array}\right]
\][/tex]
