To transform the matrix [tex]\(\begin{bmatrix} 5 & 1 & 2 \\ 2 & -2 & 6 \\ 7 & 0 & 1 \end{bmatrix}\)[/tex] into [tex]\(\begin{bmatrix} 1 & -1 & 3 \\ 0 & 6 & -13 \\ 7 & 0 & 1 \end{bmatrix}\)[/tex], follow these steps:
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.
