To find matrix [tex]\( C \)[/tex], we need to apply a series of row operations to transform matrix [tex]\( A \)[/tex] step-by-step.
### Step 1: Transform Matrix [tex]\( A \)[/tex] to Matrix [tex]\( B \)[/tex]
Matrix [tex]\( A \)[/tex] is given as:
[tex]\[
A=\begin{pmatrix}
2 & -2 & 4 & 6 \\
1 & 3 & 2 & 4 \\
2 & -1 & 4 & 6
\end{pmatrix}
\][/tex]
The first row operation is [tex]\(\frac{1}{2} R_1 \rightarrow R_1\)[/tex]:
This means we divide all elements of the first row by 2.
[tex]\[
\text{Row}_1 = \left(2, -2, 4, 6\right) \div 2 = \left(1, -1, 2, 3\right)
\][/tex]
Therefore, matrix [tex]\( B \)[/tex] becomes:
[tex]\[
B=\begin{pmatrix}
1 & -1 & 2 & 3 \\
1 & 3 & 2 & 4 \\
2 & -1 & 4 & 6
\end{pmatrix}
\][/tex]
### Step 2: Transform Matrix [tex]\( B \)[/tex] to Matrix [tex]\( C \)[/tex]
The next row operation is [tex]\(-R_1 + R_2 \rightarrow R_2\)[/tex]:
This means we subtract the first row from the second row and replace the second row with the result.
[tex]\[
\text{Row}_2 = \left(1, 3, 2, 4\right) - \left(1, -1, 2, 3\right) = \left(0, 4, 0, 1\right)
\][/tex]
Thus, matrix [tex]\( C \)[/tex] becomes:
[tex]\[
C=\begin{pmatrix}
1 & -1 & 2 & 3 \\
0 & 4 & 0 & 1 \\
2 & -1 & 4 & 6
\end{pmatrix}
\][/tex]
So, the matrix [tex]\( C \)[/tex] that represents the final transformation is:
[tex]\[
C=\begin{pmatrix}
1 & -1 & 2 & 3 \\
0 & 4 & 0 & 1 \\
2 & -1 & 4 & 6
\end{pmatrix}
\][/tex]
