Answer
The new matrix is,
$\left[ \begin{matrix}
1 & -5 & 2 & -2 & 4 \\
0 & 1 & -3 & -1 & 0 \\
0 & 15 & -4 & 5 & -6 \\
0 & -19 & 12 & -6 & 13 \\
\end{matrix} \right]$
Work Step by Step
Consider the given matrix,
$\left[ \begin{matrix}
1 & -5 & 2 & -2 & 4 \\
0 & 1 & -3 & -1 & 0 \\
3 & 0 & 2 & -1 & 6 \\
-4 & 1 & 4 & 2 & -3 \\
\end{matrix} \right]$
The operation $-3{{R}_{1}}+{{R}_{3}}$ implies that elements of the first row will be multiplied by $-3$ and then added with the corresponding elements of the third row. The operation $4{{R}_{1}}+{{R}_{4}}$ implies that elements of the first row will be multiplied by $4$ and then added with the corresponding elements of the fourth row.
The new matrix is obtained after performing the row operation ${{R}_{3}}\to -3{{R}_{1}}+{{R}_{3}},{{R}_{4}}\to 4{{R}_{1}}+{{R}_{4}}$ ,
$\left[ \begin{matrix}
1 & -5 & 2 & -2 & 4 \\
0 & 1 & -3 & -1 & 0 \\
-3+3 & 15+0 & -6+2 & 6-1 & -12+6 \\
4-4 & -20+1 & 8+4 & -8+2 & 16-3 \\
\end{matrix} \right]$
Therefore, the new matrix is
$\left[ \begin{matrix}
1 & -5 & 2 & -2 & 4 \\
0 & 1 & -3 & -1 & 0 \\
0 & 15 & -4 & 5 & -6 \\
0 & -19 & 12 & -6 & 13 \\
\end{matrix} \right]$
