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We are tasked with finding the values of \( k \) for which the given augmented matrix corresponds to a consistent linear system. Consistency means that there are either infinitely many solutions or a unique solution, but no contradictions (like having a row of zeros equated to a nonzero number).

Let's go through the problems one by one.

### Problem 19 (a):
The augmented matrix is:

\[
\begin{bmatrix}
1 & k & -4 \\
4 & 8 & 2
\end{bmatrix}
\]

We want to determine for which values of \( k \) the system is consistent. This can be done by row-reducing the matrix. I'll first perform the row reduction.

#### Step 1: Eliminate the first column of the second row.
Multiply the first row by 4 and subtract it from the second row:

\[
R_2 = R_2 - 4 \cdot R_1
\]

\[
\begin{bmatrix}
1 & k & -4 \\
4 & 8 & 2
\end{bmatrix}
\longrightarrow
\begin{bmatrix}
1 & k & -4 \\
0 & 8 - 4k & 18
\end{bmatrix}
\]

The matrix is now:

\[
\begin{bmatrix}
1 & k & -4 \\
0 & 8 - 4k & 18
\end{bmatrix}
\]

#### Step 2: Analyze consistency.
For the system to be consistent, we need the second row to not introduce contradictions. Specifically, we require that \( 8 - 4k \neq 0 \). If \( 8 - 4k = 0 \), we would have a row of zeros in the coefficient part but a nonzero constant, which would make the system inconsistent.

Solving \( 8 - 4k = 0 \):

\[
8 - 4k = 0 \quad \Rightarrow \quad k = 2
\]

Thus, the system is inconsistent when \( k = 2 \). For any other value of \( k \), the system will be consistent.

### Conclusion for 19 (a):
The system is consistent for all \( k \neq 2 \).

---

Would you like a detailed explanation for problem 19 (b), or would you like to continue with problems 20 (a) and (b)?

### Related Questions:
1. What does it mean for a linear system to have infinitely many solutions?
2. How does row reduction help determine consistency in a system?
3. What are the differences between an augmented matrix and a coefficient matrix?
4. How can we interpret the determinant of a matrix in terms of consistency?
5. Can there be inconsistent systems even if there are multiple variables?

### Tip:
When determining the consistency of a system using an augmented matrix, focus on reducing the matrix to row echelon form (REF). A row of zeros in the coefficient part but with a nonzero entry in the augmented column will signal inconsistency.

In Exercises 19–20, find all values of k for which the given augmented matrix corresponds to a consistent linear system.

Solve linear algebra exercises to find values of k that ensure the augmented matrix represents a consistent linear system. Learn step-by-step methods, including row reduction and analyzing the echelon form for consistency.

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