Answer
The new matrix, $\left[ \begin{matrix}
1 & 2 & 2 & 2 \\
0 & 1 & -1 & 2 \\
0 & 0 & 9 & -9 \\
\end{matrix} \right]$
Work Step by Step
Consider the given matrix, $\left[ \begin{matrix}
1 & 2 & 2 & 2 \\
0 & 1 & -1 & 2 \\
0 & 5 & 4 & 1 \\
\end{matrix} \right]$
Perform the operation, $-5{{R}_{2}}+{{R}_{3}}$
For this we will first multiply row 2 by $\left( -5 \right)$ to get, $\begin{align}
& 0\left( -5 \right)=0 \\
& 1\left( -5 \right)=-5 \\
& -1\left( -5 \right)=5 \\
& 2\left( -5 \right)=-10
\end{align}$
Now adding this with row 3 as below, we get:
$\left[ \begin{matrix}
1 & 2 & 2 & 2 \\
0 & 1 & -1 & 2 \\
0+0 & 5+\left( -5 \right) & 4+5 & 1+\left( -10 \right) \\
\end{matrix} \right]=\left[ \begin{matrix}
1 & 2 & 2 & 2 \\
0 & 1 & -1 & 2 \\
0 & 0 & 9 & -9 \\
\end{matrix} \right]$
The new matrix is: $\left[ \begin{matrix}
1 & 2 & 2 & 2 \\
0 & 1 & -1 & 2 \\
0 & 0 & 9 & -9 \\
\end{matrix} \right]$
