1. Scale the second row [tex]\(R_2\)[/tex] by [tex]\(\frac{1}{2}\)[/tex]: [tex]\[
\frac{1}{2} R_2
\][/tex] This changes the matrix to: [tex]\[
\begin{bmatrix}
5 & 1 & 2 \\
1 & -1 & 3 \\
7 & 0 & 1
\end{bmatrix}
\][/tex]
2. Replace the first row [tex]\(R_1\)[/tex] with [tex]\(-5 R_2 + R_1\)[/tex]: [tex]\[
-5 R_2 + R_1 \text{ replaces } R_1
\][/tex] This changes the matrix to: [tex]\[
\begin{bmatrix}
0 & 6 & -13 \\
1 & -1 & 3 \\
7 & 0 & 1
\end{bmatrix}
\][/tex]
3. Switch the first row [tex]\(R_1\)[/tex] and the second row [tex]\(R_2\)[/tex]: [tex]\[
\text{switch } R_2 \text{ and } R_1
\][/tex] This changes the matrix to: [tex]\[
\begin{bmatrix}
1 & -1 & 3 \\
0 & 6 & -13 \\
7 & 0 & 1
\end{bmatrix}
\][/tex]
So, the sequence of row operations is: 1. [tex]\(\frac{1}{2} R_2\)[/tex] 2. [tex]\(-5 R_2 + R_1 \text{ replaces } R_1\)[/tex] 3. \text{switch } [tex]\(R_2\)[/tex] and [tex]\(R_1\)[/tex]
These steps transform the initial matrix into the desired final matrix.
