Math Problem Statement
Find all the solutions of Ax = 0, where A is row equivalent to the given matrix. The free variable(s) is/are (select all that apply): x1, x2, x3.
Solution
To solve the given problem, we need to find the solutions of the homogeneous system , where the matrix is provided:
8 & 1 & -8 \\ 16 & 2 & -16 \end{bmatrix}$$ We will proceed with the following steps: ### 1. Row Reduce the Matrix: The goal is to row reduce the matrix to its row echelon form. We can perform elementary row operations to simplify it. $$\text{Initial Matrix:} \quad \begin{bmatrix} 8 & 1 & -8 \\ 16 & 2 & -16 \end{bmatrix}$$ First, we can replace row 2 ($$R_2$$) by $$R_2 - 2R_1$$: $$R_2 \rightarrow R_2 - 2R_1 = \begin{bmatrix} 16 & 2 & -16 \end{bmatrix} - 2 \begin{bmatrix} 8 & 1 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$$ The row-reduced matrix is now: $$\begin{bmatrix} 8 & 1 & -8 \\ 0 & 0 & 0 \end{bmatrix}$$ ### 2. Analyze the Row-Reduced System: This gives us the following system of linear equations: $$8x_1 + x_2 - 8x_3 = 0$$ $$0 = 0$$ The second row is trivial, so we focus on the first equation: $$8x_1 + x_2 - 8x_3 = 0$$ Solving for $$x_1$$: $$x_1 = \frac{-1}{8}x_2 + x_3$$ ### 3. Identify Free Variables: In this system, $$x_1$$ is expressed in terms of $$x_2$$ and $$x_3$$. Therefore, $$x_2$$ and $$x_3$$ are the free variables. ### Conclusion: The free variables are $$x_2$$ and $$x_3$$, meaning the correct answer is: - $$x_2$$ - $$x_3$$ Would you like more details on any part of this process? Here are some related questions for further understanding: 1. What are the steps of Gaussian elimination in more detail? 2. How do you recognize free variables in a system of linear equations? 3. Can you explain why the second row being zero leads to infinite solutions? 4. What does the row-reduced form of a matrix represent in terms of solution sets? 5. How does this solution change if the matrix had more rows or variables? **Tip**: When solving homogeneous systems like this, always look for rows of zeros, as they indicate free variables and imply infinite solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Homogeneous Systems
Gaussian Elimination
Formulas
x1 = (-1/8)x2 + x3
Theorems
Row Echelon Form
Suitable Grade Level
University Level
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