https://file.aiquickdraw.com/mathbot/file/17267560253850j6y3ngs.jpg

Let's analyze the augmented matrices in echelon form from the image.

We are given two matrices, labeled (a) and (b), and asked to determine the consistency and whether the solution is unique.

### Matrix (a):
\[
\begin{pmatrix}
* & * & * \\
0 & 0 & * \\
0 & 0 & 0
\end{pmatrix}
\]

- The last row is \((0, 0, 0)\), indicating no constraints from this row, so it doesn't affect consistency.
- The second row is \((0, 0, *)\), where the "*" indicates a non-zero constant (because the question is in augmented form). This row represents an equation of the form \(0x + 0y = \text{non-zero value}\), which is a **contradiction**.
  
  **Conclusion for (a):** The system is **inconsistent** because of the contradiction in the second row.

---

### Matrix (b):
\[
\begin{pmatrix}
* & * & * & * \\
0 & * & * & * \\
0 & 0 & 0 & *
\end{pmatrix}
\]

- The last row is \((0, 0, 0, *)\), which represents the equation \(0x + 0y + 0z = \text{non-zero value}\). This is a **contradiction** because there is no way to satisfy this equation.
  
  **Conclusion for (b):** The system is also **inconsistent** because of the contradiction in the last row.

### Summary:
- Both systems (a) and (b) are **inconsistent**.
- Since they are inconsistent, no solution exists for either system.

Would you like any further details or clarifications?

Here are five related questions:
1. What does it mean for a system to be consistent?
2. How do you determine whether a system has a unique solution?
3. What is the role of echelon form in solving systems of equations?
4. Can a system have infinitely many solutions? Under what conditions?
5. How does row reduction help in determining the consistency of a system?

**Tip:** When checking consistency, always look for contradictions in the form of rows like \((0, 0, 0, \dots , c)\), where \(c \neq 0\), as these indicate inconsistency.

Use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique.

Determine if a system of linear equations is consistent using echelon form. This problem involves analyzing matrices, identifying contradictions, and checking for unique or multiple solutions. Suitable for linear algebra students.

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