To transform the matrix [tex]\(\left[\begin{array}{ccc}5 & 1 & 2 \\ 2 & -2 & 6 \\ 7 & 0 & 1\end{array}\right]\)[/tex] to the matrix [tex]\(\left[\begin{array}{ccc}1 & -1 & 3 \\ 0 & 6 & -13 \\ 7 & 0 & 1\end{array}\right]\)[/tex], we need to perform a sequence of row operations. Here are the steps:
1. Multiply the second row by 2:
[tex]\(2 R_2 \rightarrow R_2\)[/tex]
2. Use the new second row to update the first row:
[tex]\(-5 R_2 + R_1 \rightarrow R_1\)[/tex]
Here is the detailed step-by-step process:
Start with the matrix:
[tex]\[
\left[\begin{array}{ccc}
5 & 1 & 2 \\
2 & -2 & 6 \\
7 & 0 & 1
\end{array}\right]
\][/tex]
1. Multiply the second row by 2 to get:
[tex]\[
2 \cdot \left[\begin{array}{ccc}
2 & -2 & 6
\end{array}\right] = \left[\begin{array}{ccc}
4 & -4 & 12
\end{array}\right]
\][/tex]
Now the matrix looks like this:
[tex]\[
\left[\begin{array}{ccc}
5 & 1 & 2 \\
4 & -4 & 12 \\
7 & 0 & 1
\end{array}\right]
\][/tex]
2. Update the first row by adding -5 times the new second row to the first row:
[tex]\[
\left(5, 1, 2 \right) + (-5 \cdot \left(4, -4, 12\right)) = \left(5, 1, 2\right) + \left(-20, 20, -60\right) = \left[-15, 21, -58\right]
\][/tex]
Now the matrix looks like this:
[tex]\[
\left[\begin{array}{ccc}
-15 & 21 & -58 \\
4 & -4 & 12 \\
7 & 0 & 1
\end{array}\right]
\][/tex]
Thus, the transformation sequence is:
1. [tex]\(2 R_2 \rightarrow R_2\)[/tex]
2. [tex]\(-5 R_2 + R_1 \rightarrow R_1\)[/tex]
So, the correct sequence of row operations is:
- [tex]\(2 R_2\)[/tex]
- [tex]\(-5 R_2 + R_1\)[/tex]
Fill in the boxes accordingly:
- [tex]\(2 R_2\)[/tex]
- [tex]\(-5 R_2 + R_1\)[/tex] replaces [tex]\(R_1\)[/tex]
