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 


To determine the values of \( k \) that make this system inconsistent, let's analyze the augmented matrix:

\[
\begin{pmatrix}
k & 1 & 1 & | & 0 \\
0 & k-1 & 0 & | & -2 \\
0 & 0 & k^2-1 & | & k+1 \\
\end{pmatrix}
\]

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.

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

Learn to identify values of k that make an augmented matrix system inconsistent. This problem explores key linear algebra concepts such as matrix consistency conditions and augmented matrix solutions, ideal for undergraduate students.

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