Math Problem Statement
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
[1 h -3] [-2 4 6]
Solution
To determine the value(s) of such that the given augmented matrix corresponds to a consistent linear system, we first need to understand the structure of the augmented matrix.
The augmented matrix is:
1 & h & -3 \\ -2 & 4 & 6 \end{array}\right]$$ This matrix represents a system of two linear equations in three variables. To determine when the system is consistent (i.e., has at least one solution), we need to analyze the system through Gaussian elimination or row reduction. ### Step 1: Row Reduction We perform row operations to reduce this matrix to row echelon form. Start with the matrix: $$\left[\begin{array}{ccc|c} 1 & h & -3 & 0 \\ -2 & 4 & 6 & 0 \end{array}\right]$$ We aim to eliminate the $$-2$$ in the second row, first column. To do this, add 2 times the first row to the second row: $$R_2 \rightarrow R_2 + 2R_1$$ $$\left[\begin{array}{ccc|c} 1 & h & -3 & 0 \\ 0 & 4+2h & 6-6 & 0 \end{array}\right]$$ This simplifies to: $$\left[\begin{array}{ccc|c} 1 & h & -3 & 0 \\ 0 & 4+2h & 0 & 0 \end{array}\right]$$ ### Step 2: Analyze the Second Row For the system to be consistent, the second row must not represent a contradiction. In other words, we need the second row to not be of the form $$0 \ 0 \ 0 | \text{non-zero constant}$$, which would imply no solution. The second row is: $$0 \ 4+2h \ 0 | 0$$ For the system to be consistent, we must ensure that the coefficient of $$y$$ (the second variable) in the second row, $$4 + 2h$$, is not zero. If $$4 + 2h = 0$$, the second row would imply $$0 = 0$$, which would leave the system undetermined and potentially inconsistent depending on the other equations. ### Step 3: Solve for $$h$$ Set $$4 + 2h = 0$$ and solve for $$h$$: $$4 + 2h = 0$$ $$2h = -4$$ $$h = -2$$ ### Step 4: Conclusion For $$h = -2$$, the second row becomes $$0 \ 0 \ 0 | 0$$, which does not provide new information and still allows the system to be consistent. Therefore, **the system is consistent for all values of $$h$$**. ### Summary: The system is consistent for **all values of $$h$$**. Would you like further details or have any other questions on this? ### Related Questions: 1. What is the interpretation of an augmented matrix corresponding to a consistent system? 2. How does Gaussian elimination help in solving linear systems? 3. What does it mean for a system of equations to be inconsistent? 4. How do you determine the rank of a matrix and its implications for consistency? 5. Can a system of equations with more variables than equations still be consistent? ### Tip: When performing row operations, always check for rows that turn into all zeros, as this can provide valuable information about the consistency and rank of the system.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Consistency of Linear Systems
Formulas
Row reduction to echelon form
Consistency condition for linear systems
Theorems
Consistency criterion for systems of linear equations
Suitable Grade Level
Grades 9-12
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