Answer
The new matrix is, $\left[ \begin{matrix}
1 & -1 & \frac{1}{2} & -\frac{1}{2} \\
1 & 2 & -1 & 2 \\
6 & 4 & 3 & 5 \\
\end{matrix} \right]$
Work Step by Step
Consider the given matrix, $\left[ \begin{matrix}
2 & -2 & 1 & -1 \\
1 & 2 & -1 & 2 \\
6 & 4 & 3 & 5 \\
\end{matrix} \right]$
Perform the operation, $\frac{1}{2}{{R}_{1}}$
For this we will multiply elements of row 1 by $\left( \frac{1}{2} \right)$ to get the matrix as below:
$\left[ \begin{matrix}
2\times \frac{1}{2} & -2\times \frac{1}{2} & 1\times \frac{1}{2} & -1\times \frac{1}{2} \\
1 & 2 & -1 & 2 \\
6 & 4 & 3 & 5 \\
\end{matrix} \right]=\left[ \begin{matrix}
1 & -1 & \frac{1}{2} & -\frac{1}{2} \\
1 & 2 & -1 & 2 \\
6 & 4 & 3 & 5 \\
\end{matrix} \right]$
The new matrix is: $\left[ \begin{matrix}
1 & -1 & \frac{1}{2} & -\frac{1}{2} \\
1 & 2 & -1 & 2 \\
6 & 4 & 3 & 5 \\
\end{matrix} \right]$.
