Math Problem Statement
Solve the linear system 𝐴𝐱=𝐛Ax=b using Gaussian elimination.
𝐴=⎡⎣⎢⎢210301411⎤⎦⎥⎥𝐛=⎡⎣⎢⎢100⎤⎦⎥⎥.A=[234101011]b=[100].
Enter your answers below: The entries of 𝐱=[𝑥1𝑥2𝑥3]𝑇x=[x1x2x3]T are
𝑥1=x1= Answer 1 Question 4
𝑥2=x2= Answer 2 Question 4
𝑥3=x3= Answer 3 Question 4
Solution
We will solve the linear system using Gaussian elimination.
Given system:
\mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$ #### **Step 1: Augmented Matrix** $$[A | \mathbf{b}] = \begin{bmatrix} 2 & 3 & 4 & | 1 \\ 1 & 0 & 1 & | 0 \\ 0 & 1 & 1 & | 0 \end{bmatrix}$$ #### **Step 2: Forward Elimination** - Swap **row 1** and **row 2** to make the first pivot 1: $$\begin{bmatrix} 1 & 0 & 1 & | 0 \\ 2 & 3 & 4 & | 1 \\ 0 & 1 & 1 & | 0 \end{bmatrix}$$ - Eliminate first column below pivot: - Row 2 → Row 2 - 2 × Row 1: $$\begin{bmatrix} 1 & 0 & 1 & | 0 \\ 0 & 3 & 2 & | 1 \\ 0 & 1 & 1 & | 0 \end{bmatrix}$$ - Make second pivot 1 by dividing row 2 by 3: $$\begin{bmatrix} 1 & 0 & 1 & | 0 \\ 0 & 1 & \frac{2}{3} & | \frac{1}{3} \\ 0 & 1 & 1 & | 0 \end{bmatrix}$$ - Eliminate second column below pivot: - Row 3 → Row 3 - Row 2: $$\begin{bmatrix} 1 & 0 & 1 & | 0 \\ 0 & 1 & \frac{2}{3} & | \frac{1}{3} \\ 0 & 0 & \frac{1}{3} & | -\frac{1}{3} \end{bmatrix}$$ - Make third pivot 1 by multiplying row 3 by 3: $$\begin{bmatrix} 1 & 0 & 1 & | 0 \\ 0 & 1 & \frac{2}{3} & | \frac{1}{3} \\ 0 & 0 & 1 & | -1 \end{bmatrix}$$ #### **Step 3: Back Substitution** From the third row: $$x_3 = -1$$ From the second row: $$x_2 + \frac{2}{3}(-1) = \frac{1}{3}$$ $$x_2 - \frac{2}{3} = \frac{1}{3}$$ $$x_2 = 1$$ From the first row: $$x_1 + (-1) = 0$$ $$x_1 = 1$$ ### **Final Solution** $$\mathbf{x} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$$ **Answers:** - $$x_1 = 1$$ - $$x_2 = 1$$ - $$x_3 = -1$$ --- Do you want a more detailed explanation or any clarifications? Here are 5 related questions to expand your understanding: 1. How does Gaussian elimination handle inconsistent or dependent systems? 2. What is the difference between Gaussian and Gauss-Jordan elimination? 3. How can row swapping affect numerical stability in Gaussian elimination? 4. How does back substitution work in solving triangular systems? 5. What are the computational complexities of Gaussian elimination? **Tip:** Always check the determinant of $$A$$ to verify if a unique solution exists!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Matrix Operations
Formulas
Gaussian elimination involves forward elimination to convert the matrix to row echelon form, followed by back substitution to find the solution.
Theorems
Gaussian Elimination Method
Suitable Grade Level
Undergraduate (Linear Algebra Course)
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