To determine which matrix represents matrix [tex]\( C \)[/tex], let's go through the transformations step-by-step.
1. Starting with Matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{array}{ccc|c}
2 & -2 & 4 & 6 \\
1 & 3 & 2 & 4 \\
2 & -1 & 4 & 6
\end{array} \][/tex]
2. First Transformation: [tex]\(\frac{1}{2} R1 \rightarrow R1\)[/tex]
- Divide the first row by 2.
[tex]\[
B = \begin{array}{ccc|c}
1 & -1 & 2 & 3 \\
1 & 3 & 2 & 4 \\
2 & -1 & 4 & 6
\end{array}
\][/tex]
3. Second Transformation: [tex]\(-R1 + R2 \rightarrow R2\)[/tex]
- Subtract the first row from the second row.
[tex]\[
R2 \rightarrow R2 - R1 = \begin{array}{ccc|c}
1 & -1 & 2 & 3 \\
(1 - 1) & (3 - (-1)) & (2 - 2) & (4 - 3) \\
2 & -1 & 4 & 6
\end{array} = \begin{array}{ccc|c}
1 & -1 & 2 & 3 \\
0 & 4 & 0 & 1 \\
2 & -1 & 4 & 6
\end{array}
\][/tex]
Thus, the resulting matrix [tex]\( C \)[/tex] is:
[tex]\[
C = \begin{array}{ccc|c}
1 & -1 & 2 & 3 \\
0 & 4 & 0 & 1 \\
2 & -1 & 4 & 6
\end{array}
\][/tex]
The correct matrix representing matrix [tex]\( C \)[/tex] is:
[tex]\[
\left.\left\lvert\, \begin{array}{ccc|c}
1 & -1 & 2 & 3 \\
0 & 4 & 0 & 1 \\
2 & -1 & 4 & 6
\end{array}\right.\right]
\][/tex]
