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To solve this problem, we need to determine the value(s) of \( h \) such that the given matrix represents the augmented matrix of a consistent linear system.

The matrix provided is:

\[
\begin{bmatrix}
1 & -1 & 4 \\
-2 & 3 & h
\end{bmatrix}
\]

This is the augmented matrix of a system of two linear equations. For the system to be consistent, the system should have either a unique solution or infinitely many solutions, meaning the equations should not contradict each other.

### Step 1: Perform Row Reduction (Gaussian Elimination)
We will reduce the matrix to row echelon form to see if the system remains consistent for all values of \( h \).

#### Matrix:

\[
\begin{bmatrix}
1 & -1 & 4 \\
-2 & 3 & h
\end{bmatrix}
\]

#### Row 2 Transformation
We can eliminate the first element of the second row by adding \( 2 \times \) (Row 1) to Row 2:

\[
\text{Row 2} \rightarrow \text{Row 2} + 2 \times \text{Row 1}
\]

\[
\begin{aligned}
R_2: & \quad -2 + 2(1) = 0 \\
& \quad 3 + 2(-1) = 1 \\
& \quad h + 2(4) = h + 8
\end{aligned}
\]

So, the transformed matrix is:

\[
\begin{bmatrix}
1 & -1 & 4 \\
0 & 1 & h + 8
\end{bmatrix}
\]

### Step 2: Interpret the Result
The system of equations corresponding to this matrix is:

1. \( x - y = 4 \)
2. \( y = h + 8 \)

This system is consistent for **all values of \( h \)** because there are no contradictions between the two equations. For any value of \( h \), we can solve for \( x \) and \( y \).

Thus, the matrix is consistent for any value of \( h \).

### Conclusion
The matrix represents a consistent linear system for all values of \( h \).

---

Would you like further details or clarifications on any part of the solution? 

Here are some related questions you might find useful:
1. How is Gaussian elimination used to solve linear systems?
2. What is the difference between a consistent and an inconsistent system?
3. How do you interpret row echelon form in terms of solutions to a system?
4. What conditions lead to a system having infinitely many solutions?
5. Can a system have no solution? If yes, how is that determined?

**Tip:** When performing Gaussian elimination, always focus on simplifying the matrix step by step, aiming to make it as easy as possible to solve the system.

In Exercises 17 and 18, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. [ 1 -1 4; -2 3 h ]

Find the value of h such that the augmented matrix [1 -1 4; -2 3 h] represents a consistent linear system. This solution involves Gaussian elimination, row reduction, and interpretation of row echelon form.

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