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The problem asks to analyze the compatibility of a system of linear equations \( Ax = b \), where:

\[
A = \begin{bmatrix} 
1 & m-1 & 2 \\
m & 0 & -1 \\
4 & m-1 & 2 
\end{bmatrix}
\quad \text{and} \quad 
b = \begin{bmatrix} 
1 \\
1 \\
2 
\end{bmatrix}
\]

It requests the use of elementary transformations and the **Rouché-Frobenius theorem** to study the compatibility based on the parameter \( m \in \mathbb{R} \).

### Steps:

1. **Apply the Rouché-Frobenius Theorem**: This theorem states that a system of linear equations \( Ax = b \) has a solution if and only if the rank of the matrix \( A \) equals the rank of the augmented matrix \( [A | b] \). Furthermore, if the number of unknowns is equal to the rank, the system has a unique solution; otherwise, it may have infinitely many solutions.

2. **Determine the ranks of \( A \) and \( [A|b] \)**:
    - We need to find the rank of the coefficient matrix \( A \) and compare it to the rank of the augmented matrix \( [A | b] \).
    - The ranks depend on the value of the parameter \( m \).

3. **Elementary row transformations**: 
    - To simplify \( A \) and \( [A|b] \), we will perform elementary row operations (like Gaussian elimination) and express the system in row-echelon form to calculate the ranks for different values of \( m \).

Let me start calculating these transformations and the conditions for compatibility based on \( m \). I'll go ahead with the row-reduction process to analyze the system step by step.The row-reduced form (RREF) of matrix \( A \) is:

\[
A_{rref} = \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\]

And the RREF of the augmented matrix \( [A | b] \) is:

\[
[A | b]_{rref} = \begin{bmatrix} 
1 & 0 & 0 & \frac{1}{3} \\
0 & 1 & 0 & \frac{8 - 2m}{3(m - 1)} \\
0 & 0 & 1 & \frac{m}{3} - 1 
\end{bmatrix}
\]

### Analysis of the system:
1. **Rank of \( A \)**: The matrix \( A \) is of full rank (3), meaning that for most values of \( m \), the rank is 3.
2. **Rank of \( [A | b] \)**: The rank of the augmented matrix depends on the values of \( m \). For the system to be compatible, we must have:

   \[
   \frac{8 - 2m}{3(m - 1)} \quad \text{and} \quad \frac{m}{3} - 1 \quad \text{both defined}.
   \]

   - The expression \( \frac{8 - 2m}{3(m - 1)} \) is undefined when \( m = 1 \).
   - For \( m = 1 \), the second row would contain zeros, reducing the rank of the augmented matrix.

### Conclusion:
- When \( m \neq 1 \), the system is compatible, and a unique solution exists because both \( \text{rank}(A) = \text{rank}([A | b]) = 3 \).
- When \( m = 1 \), the system becomes inconsistent since the augmented matrix's rank becomes less than 3, violating the Rouché-Frobenius theorem.

Would you like further details on any specific step? 

Here are 5 related questions for further exploration:
1. What happens when \( m = 1 \) in more detail?
2. How can we solve the system for specific values of \( m \)?
3. How would the solution vary if the vector \( b \) were different?
4. What is the geometric interpretation of the rank condition in this problem?
5. How can we check the consistency of other linear systems using the Rouché-Frobenius theorem?

**Tip**: To analyze a system of linear equations, always check the ranks of both the coefficient and augmented matrices to determine compatibility.

Considere el sistema de ecuaciones lineales Ax = b, donde A = [[1, m - 1, 2], [m, 0, -1], [4, m - 1, 2]] y b = [[1], [1], [2]]. Usando el teorema de Rouché-Frobenius, estudiar la compatibilidad del sistema en función del parámetro m ∈ R.

Analyze the compatibility of a system of linear equations using the matrix A = [[1, m - 1, 2], [m, 0, -1], [4, m - 1, 2]] and vector b = [[1], [1], [2]] based on the parameter m. Apply the Rouché-Frobenius Theorem to determine the solution and compatibility conditions.

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