To solve this, we will perform the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex] and determine the resulting matrix.
Let's start with the initial matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 6 \\
1 & 1 & 1 & -2 \\
0 & 2 & 1 & 5
\end{array}\right]
\][/tex]
We need to apply the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex]. This means we will subtract the elements of row 1 from the corresponding elements of row 2 to get the new row 2.
Step-by-step, it looks like this:
1. Element in column 1 of R2:
[tex]\[
1 - 1 = 0
\][/tex]
2. Element in column 2 of R2:
[tex]\[
1 - 2 = -1
\][/tex]
3. Element in column 3 of R2:
[tex]\[
1 - 3 = -2
\][/tex]
4. Element in the augmented column of R2:
[tex]\[
-2 - 6 = -8
\][/tex]
The updated matrix is then:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 6 \\
0 & -1 & -2 & -8 \\
0 & 2 & 1 & 5
\end{array}\right]
\][/tex]
Therefore, the result of the operation [tex]\(-R1 + R2 \rightarrow R2\)[/tex] applied to the given matrix is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 6 \\
0 & -1 & -2 & -8 \\
0 & 2 & 1 & 5
\end{array}\right]
\][/tex]
Among the given options, the correct resulting matrix is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 6 \\
0 & -1 & -2 & -8 \\
0 & 2 & 1 & 5
\end{array}\right]
\][/tex]
So, the correct option is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 2 & 3 & 6 \\
0 & -1 & -2 & -8 \\
0 & 2 & 1 & 5
\end{array}\right]
\][/tex]
