To solve the system of linear equations using Gaussian elimination, follow these steps carefully. We'll transform the augmented matrix step-by-step to reach its row-echelon form:
Given the augmented matrix:
[tex]\[
\begin{bmatrix}
1 & 2 & -3 & 3 \\
2 & -1 & -2 & 14 \\
3 & -1 & 5 & -5
\end{bmatrix}
\][/tex]
### Step 1: Eliminate the first element of the second and third rows
1. Make the first element of the second row zero:
[tex]\[
R2 \rightarrow R2 - 2 \cdot R1
\][/tex]
[tex]\[
R2 = [2, -1, -2, 14] - 2 \cdot [1, 2, -3, 3]
\][/tex]
[tex]\[
R2 = [2 - 2 \cdot 1, -1 - 2 \cdot 2, -2 + 2 \cdot 3, 14 - 2 \cdot 3]
\][/tex]
[tex]\[
R2 = [0, -5, 4, 8]
\][/tex]
Our matrix now looks like this:
[tex]\[
\begin{bmatrix}
1 & 2 & -3 & 3 \\
0 & -5 & 4 & 8 \\
3 & -1 & 5 & -5
\end{bmatrix}
\][/tex]
2. Make the first element of the third row zero:
[tex]\[
R3 \rightarrow R3 - 3 \cdot R1
\][/tex]
[tex]\[
R3 = [3, -1, 5, -5] - 3 \cdot [1, 2, -3, 3]
\][/tex]
[tex]\[
R3 = [3 - 3 \cdot 1, -1 - 3 \cdot 2, 5 + 3 \cdot 3, -5 - 3 \cdot 3]
\][/tex]
[tex]\[
R3 = [0, -7, 14, -14]
\][/tex]
Our matrix now looks like this:
[tex]\[
\begin{bmatrix}
1 & 2 & -3 & 3 \\
0 & -5 & 4 & 8 \\
0 & -7 & 14 & -14
\end{bmatrix}
\][/tex]
### Step 2: Make the second element of the second row one
To make the second element of the second row one:
[tex]\[
R2 \rightarrow \frac{R2}{-5}
\][/tex]
[tex]\[
R2 = \frac{1}{-5} \cdot [0, -5, 4, 8]
\][/tex]
[tex]\[
R2 = [0, 1, -0.8, -1.6]
\][/tex]
Our matrix now looks like this:
[tex]\[
\begin{bmatrix}
1 & 2 & -3 & 3 \\
0 & 1 & -0.8 & -1.6 \\
0 & -7 & 14 & -14
\end{bmatrix}
\][/tex]
### Step 3: Eliminate the second element of the third row
To make the second element of the third row zero:
[tex]\[
R3 \rightarrow R3 + 7 \cdot R2
\][/tex]
[tex]\[
R3 = [0, -7, 14, -14] + 7 \cdot [0, 1, -0.8, -1.6]
\][/tex]
[tex]\[
R3 = [0 + 7 \cdot 0, -7 + 7 \cdot 1, 14 + 7 \cdot (-0.8), -14 + 7 \cdot (-1.6)]
\][/tex]
[tex]\[
R3 = [0, 0, 14 - 5.6, -14 - 11.2]
\][/tex]
[tex]\[
R3 = [0, 0, 8.4, -25.2]
\][/tex]
Our final row-echelon form of the augmented matrix is:
[tex]\[
\begin{bmatrix}
1 & 2 & -3 & 3 \\
0 & 1 & -0.8 & -1.6 \\
0 & 0 & 8.4 & -25.2
\end{bmatrix}
\][/tex]
