To transform the initial matrix [tex]\( A = \left[\begin{array}{ccc} 2 & 3 & 1 \\ 0 & -4 & 5 \\ 1 & 5 & -7 \end{array}\right] \)[/tex] into the matrix [tex]\( \left[\begin{array}{ccc} 3 & 8 & -6 \\ 4 & 2 & 7 \\ 2 & 3 & 1 \end{array}\right] \)[/tex], the following sequence of row operations should be applied:
1. Switch [tex]\(R_1\)[/tex] and [tex]\(R_3\)[/tex]. This operation will swap the first row and the third row:
[tex]\[
\left[\begin{array}{ccc}
1 & 5 & -7 \\
0 & -4 & 5 \\
2 & 3 & 1
\end{array}\right]
\][/tex]
2. Replace [tex]\(R_2\)[/tex] with [tex]\(R_2 + 2R_3\)[/tex]. This operation will add 2 times the third row to the second row:
[tex]\[
\left[\begin{array}{ccc}
1 & 5 & -7 \\
0 + 2 \cdot 2 & -4 + 2 \cdot 3 & 5 + 2 \cdot 1 \\
2 & 3 & 1
\end{array}\right]
= \left[\begin{array}{ccc}
1 & 5 & -7 \\
4 & 2 & 7 \\
2 & 3 & 1
\end{array}\right]
\][/tex]
3. Replace [tex]\(R_1\)[/tex] with [tex]\(R_1 + R_3\)[/tex]. This operation will add the third row to the first row:
[tex]\[
\left[\begin{array}{ccc}
1 + 2 & 5 + 3 & -7 + 1 \\
4 & 2 & 7 \\
2 & 3 & 1
\end{array}\right]
= \left[\begin{array}{ccc}
3 & 8 & -6 \\
4 & 2 & 7 \\
2 & 3 & 1
\end{array}\right]
\][/tex]
Therefore, these row operations successfully transform the given matrix [tex]\( A \)[/tex] into the desired matrix. The sequence of operations is:
1. Switch [tex]\(R_1\)[/tex] and [tex]\(R_3\)[/tex].
2. [tex]\( R_2 + 2R_3 \)[/tex] replaces [tex]\( R_2 \)[/tex].
3. [tex]\( R_1 + R_3 \)[/tex] replaces [tex]\( R_1 \)[/tex].
