To perform the elementary row operation [tex]\(2 R_1 + R_3 \rightarrow R_3\)[/tex] on the given matrix, we need to follow these steps:
Given matrix:
[tex]\[
\left[\begin{array}{ccc:c}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
3 & 1 & -3 & 5
\end{array}\right]
\][/tex]
### Step 1: Identify the rows
- [tex]\( R_1 = [2, 3, 5, 2] \)[/tex]
- [tex]\( R_3 = [3, 1, -3, 5] \)[/tex]
### Step 2: Scale [tex]\( R_1 \)[/tex] by 2
- [tex]\( 2 R_1 = [2 \times 2, 2 \times 3, 2 \times 5, 2 \times 2] = [4, 6, 10, 4] \)[/tex]
### Step 3: Add the scaled [tex]\( R_1 \)[/tex] to [tex]\( R_3 \)[/tex]:
- [tex]\( R_3 \leftarrow 2 R_1 + R_3 \)[/tex]
Let's perform the addition element-wise:
- First element: [tex]\( 4 + 3 = 7 \)[/tex]
- Second element: [tex]\( 6 + 1 = 7 \)[/tex]
- Third element: [tex]\( 10 + (-3) = 7 \)[/tex]
- Fourth element: [tex]\( 4 + 5 = 9 \)[/tex]
So, the updated [tex]\( R_3 \)[/tex] is:
[tex]\[
R_3 = [7, 7, 7, 9]
\][/tex]
### Step 4: Update the matrix with the new [tex]\( R_3 \)[/tex]:
The resulting matrix after the operation is:
[tex]\[
\left[\begin{array}{ccc:c}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
7 & 7 & 7 & 9
\end{array}\right]
\][/tex]
Thus, the matrix after performing the elementary row operation [tex]\(2 R_1 + R_3 \rightarrow R_3\)[/tex] is:
[tex]\[
\left[\begin{array}{ccc:c}
2 & 3 & 5 & 2 \\
1 & 4 & 5 & 5 \\
7 & 7 & 7 & 9
\end{array}\right]
\][/tex]
