Perform the elementary row operation:
[tex]\[ 5R_2 + R_1 \rightarrow R_1 \][/tex]

[tex]\[
\left[
\begin{array}{cc|c}
0 & -6 & 2 \\
0 & -5 & 8
\end{array}
\right]
\][/tex]



Answer :

To solve the problem of performing the elementary row operation [tex]\( 5R_2 + R_1 \rightarrow R_1 \)[/tex] on the matrix

[tex]\[ \left[\begin{array}{ccc} 0 & -6 & 2 \\ 0 & -5 & 8 \end{array}\right] \][/tex]

we will follow a step-by-step approach.

### Step 1: Identify the Multiples
First, we need to determine the multiples of the elements of the second row [tex]\( R_2 \)[/tex].

[tex]\[ 5 \times R_2 = 5 \times \left[\begin{array}{ccc} 0 & -5 & 8 \end{array}\right] = \left[\begin{array}{ccc} 0 & -25 & 40 \end{array}\right] \][/tex]

### Step 2: Add the Result to the First Row
Next, we add the result from step 1 to the elements of the first row [tex]\( R_1 \)[/tex]:

[tex]\[ R_1 + 5R_2 = \left[\begin{array}{ccc} 0 & -6 & 2 \end{array}\right] + \left[\begin{array}{ccc} 0 & -25 & 40 \end{array}\right] = \left[\begin{array}{ccc} 0 & -31 & 42 \end{array}\right] \][/tex]

### Step 3: Replace the First Row with the Result
Finally, we replace the first row [tex]\( R_1 \)[/tex] with the result from step 2:

[tex]\[ \left[\begin{array}{ccc} 0 & -31 & 42 \\ 0 & -5 & 8 \end{array}\right] \][/tex]

### Final Matrix Configuration
The matrix after performing the elementary row operation [tex]\( 5R_2 + R_1 \rightarrow R_1 \)[/tex] is:

[tex]\[ \left[\begin{array}{ccc} 0 & -31 & 42 \\ 0 & -5 & 8 \end{array}\right] \][/tex]

Thus, the resulting rows are:
- [tex]\(R_1 = [0, -31, 42] \)[/tex]
- [tex]\(R_2 = [0, -5, 8]\)[/tex]

This is the final answer.