To perform the elementary row operation [tex]\(2R_1 + R_3 \rightarrow R_3\)[/tex] on the given matrix
[tex]\[
\begin{pmatrix}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
3 & 1 & -3 & 5
\end{pmatrix}
\][/tex]
proceed as follows:
1. Identify the rows involved:
- [tex]\( R_1 \)[/tex] is the first row: [tex]\( \begin{pmatrix} 2 & 3 & 5 & 2 \end{pmatrix} \)[/tex]
- [tex]\( R_3 \)[/tex] is the third row: [tex]\( \begin{pmatrix} 3 & 1 & -3 & 5 \end{pmatrix} \)[/tex]
2. Multiply row [tex]\( R_1 \)[/tex] by 2:
[tex]\[
2R_1 = 2 \begin{pmatrix} 2 & 3 & 5 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 6 & 10 & 4 \end{pmatrix}
\][/tex]
3. Add the result of [tex]\( 2R_1 \)[/tex] to [tex]\( R_3 \)[/tex]:
[tex]\[
R_3 = \begin{pmatrix} 3 & 1 & -3 & 5 \end{pmatrix} + \begin{pmatrix} 4 & 6 & 10 & 4 \end{pmatrix} = \begin{pmatrix} 7 & 7 & 7 & 9 \end{pmatrix}
\][/tex]
4. Replace the third row with the result:
[tex]\[
\begin{pmatrix}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
7 & 7 & 7 & 9
\end{pmatrix}
\][/tex]
Thus, the matrix after performing the operation [tex]\(2R_1 + R_3 \rightarrow R_3\)[/tex] is:
[tex]\[
\begin{pmatrix}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
7 & 7 & 7 & 9
\end{pmatrix}
\][/tex]
