Math Problem Statement
Find for what values of 𝜆 and 𝜇 the system of linear equations 𝑥 + 𝑦 + 𝑧 = 2, 𝑥 + 2𝑦 + 3𝑧 = 5, 𝑥 + 3𝑦 + 𝜆𝑧 = 𝜇 has I. a unique solution II. no solution III. infinite solutions.
Solution
We analyze the given system of linear equations:
1. \quad & x + y + z = 2, \\ 2. \quad & x + 2y + 3z = 5, \\ 3. \quad & x + 3y + \lambda z = \mu. \end{aligned}$$ This system can be represented in augmented matrix form: $$\begin{bmatrix} 1 & 1 & 1 & 2 \\ 1 & 2 & 3 & 5 \\ 1 & 3 & \lambda & \mu \end{bmatrix}.$$ The solution type depends on the rank of the coefficient matrix and the augmented matrix. To find the ranks, we perform row reduction (Gaussian elimination). --- ### Step 1: Subtract row 1 from rows 2 and 3 $$\begin{bmatrix} 1 & 1 & 1 & 2 \\ 0 & 1 & 2 & 3 \\ 0 & 2 & \lambda - 1 & \mu - 2 \end{bmatrix}.$$ --- ### Step 2: Subtract twice row 2 from row 3 $$\begin{bmatrix} 1 & 1 & 1 & 2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & \lambda - 5 & \mu - 8 \end{bmatrix}.$$ The coefficient matrix is now: $$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & \lambda - 5 \end{bmatrix}.$$ --- ### Step 3: Analyze cases based on $$\lambda - 5$$ 1. **Unique Solution:** For a unique solution, the coefficient matrix must have full rank (rank 3). This requires: $$\lambda - 5 \neq 0 \quad \Rightarrow \quad \lambda \neq 5.$$ Since the system is consistent, $$\mu$$ can take any value when $$\lambda \neq 5$$. 2. **No Solution:** For no solution, the coefficient matrix must have rank 2, but the augmented matrix must have rank 3. This occurs if: $$\lambda = 5 \quad \text{and} \quad \mu - 8 \neq 0 \quad \Rightarrow \quad \lambda = 5, \, \mu \neq 8.$$ 3. **Infinite Solutions:** For infinitely many solutions, both the coefficient matrix and augmented matrix must have rank 2. This occurs if: $$\lambda = 5 \quad \text{and} \quad \mu = 8.$$ --- ### Final Answer: I. **Unique solution:** $$\lambda \neq 5$$. II. **No solution:** $$\lambda = 5$$ and $$\mu \neq 8$$. III. **Infinite solutions:** $$\lambda = 5$$ and $$\mu = 8$$. --- Would you like a step-by-step explanation of the row reduction? Here are some questions to deepen your understanding: 1. What is the significance of the rank in determining the type of solution? 2. How do augmented matrices help identify inconsistency in a system? 3. Could we visualize this problem geometrically in terms of planes? 4. Why does $$\lambda = 5$$ reduce the rank of the coefficient matrix? 5. How can determinant analysis provide an alternative approach? **Tip:** Always ensure your system is consistent before concluding infinite solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Linear Equations
Matrix Representation
Gaussian Elimination
Rank of a Matrix
Formulas
Augmented Matrix Form
Rank of Coefficient Matrix
Rank of Augmented Matrix
Theorems
Rank-Nullity Theorem
Consistency Conditions for Linear Systems
Suitable Grade Level
Undergraduate (Introductory Linear Algebra)
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