Math Problem Statement

Consider the system given by the augment system below K 1 1 0 0 k-1 0 -2 0 0 k^2-1 k+1 The system is inconsistent for

Solution

To determine the values of kk that make this system inconsistent, let's analyze the augmented matrix:

k & 1 & 1 & | & 0 \\ 0 & k-1 & 0 & | & -2 \\ 0 & 0 & k^2-1 & | & k+1 \\ \end{pmatrix}$$ The system will be inconsistent if any row implies a contradiction, such as having a row where all coefficients are zero but the augmented part is non-zero. Let's examine the rows in terms of $$k$$: ### Step 1: Check for potential contradictions 1. **Row 2**: $$0 \cdot x + (k-1) \cdot y + 0 \cdot z = -2$$ - This row gives us $$(k-1)y = -2$$. - This equation has a solution for $$y$$ unless $$k-1 = 0$$ (i.e., $$k = 1$$). 2. **Row 3**: $$0 \cdot x + 0 \cdot y + (k^2 - 1) \cdot z = k+1$$ - This row implies $$(k^2 - 1)z = k + 1$$. - For this equation to have a solution, $$k^2 - 1 \neq 0$$ (otherwise, we get a contradiction if $$k+1 \neq 0$$). ### Step 2: Identify cases that make the system inconsistent - For **Row 2** to lead to a contradiction, let $$k = 1$$: - If $$k = 1$$, then $$k-1 = 0$$, so Row 2 becomes $$0 \cdot y = -2$$, which is a contradiction. Thus, the system is inconsistent when $$k = 1$$. - For **Row 3** to lead to a contradiction: - If $$k^2 - 1 = 0$$, then $$k = \pm 1$$. - If $$k = -1$$, Row 3 becomes $$0 \cdot z = 0$$, which is consistent. However, if $$k = 1$$, we already have an inconsistency in Row 2. ### Conclusion The system is inconsistent when $$k = 1$$. Would you like more details on how each row was analyzed, or have other questions? 1. What would happen if $$k = -1$$? 2. Can this system be consistent for any value of $$k$$? 3. How do we handle augmented matrices when checking for consistency? 4. What role does each row play in determining consistency? 5. How can systems be checked for consistency in larger matrices? **Tip:** To check a system's consistency, look for any row in the form $$0 = \text{non-zero constant}$$, which signals a contradiction.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Consistency
Augmented Matrix

Formulas

Conditions for matrix consistency

Theorems

Consistency condition for augmented matrices

Suitable Grade Level

Undergraduate