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The problem presents a linear system \( Ax = b \), where the matrix \( A \) and the vector \( b \) are defined as follows:

\[
A = \begin{bmatrix}
1 & m - 1 & 2 \\
m & 0 & -1 \\
4 & m - 1 & 2
\end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \quad m \in \mathbb{R}
\]

### Task:
You are asked to study the compatibility of this system using elementary transformations and **Rouché–Frobenius theorem**, as a function of the parameter \( m \in \mathbb{R} \).

#### Steps to approach the problem:
1. **Matrix rank**: Apply elementary transformations to the augmented matrix \( [A|b] \) to simplify it.
2. **Rouché-Frobenius theorem**: This theorem states that the system \( Ax = b \) is compatible (i.e., has at least one solution) if and only if the rank of the coefficient matrix \( A \) is equal to the rank of the augmented matrix \( [A|b] \).

We will perform Gaussian elimination to check the ranks of the matrix \( A \) and the augmented matrix \( [A|b] \) for different values of \( m \).

Let's proceed with these calculations and then interpret the compatibility based on the value of \( m \). I will perform the row reduction to examine the ranks.For the specific value of \( m = 0 \), both the coefficient matrix \( A \) and the augmented matrix \( [A|b] \) have rank 3. According to the **Rouché–Frobenius theorem**, since the rank of \( A \) equals the rank of \( [A|b] \), the system is compatible (it has at least one solution).

However, we need to analyze how the system behaves for different values of \( m \) and determine if the rank changes, potentially leading to incompatible cases.

Let me examine the system for various values of \( m \) and identify any critical points where the rank changes.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, here’s what you can do manually to explore the compatibility of the system for different values of \( m \):

1. **Calculate the rank of matrix \( A \) and the augmented matrix** for each value of \( m \).
2. **Check the rank condition**: If the rank of \( A \) is equal to the rank of \( [A|b] \), the system is compatible (has a solution). If the rank of \( A \) is less than the rank of \( [A|b] \), the system is incompatible (no solution).

If you would like further assistance with specific values of \( m \), feel free to let me know!

Would you like me to clarify any steps, or would you prefer to explore another aspect of the problem? Here are some related questions you might consider:

1. How does the Rouché-Frobenius theorem apply to systems with infinitely many solutions?
2. What is the geometric interpretation of the ranks of matrices in a system of linear equations?
3. How do elementary transformations help simplify the analysis of linear systems?
4. Can the matrix \( A \) become singular (non-invertible) for some values of \( m \)?
5. What happens when the system is underdetermined (more unknowns than equations)?

**Tip**: To simplify row reduction, try focusing on one row operation at a time and work systematically to eliminate variables!

Consider the system of linear equations Ax = b, where the matrix A and vector b are given as: A = [[1, m-1, 2], [m, 0, -1], [4, m-1, 2]], b = [1, 1, 2], with m belonging to real numbers. Use elementary transformations and the Rouché-Frobenius theorem to study the compatibility of the system of linear equations based on the parameter m.

Explore the compatibility of a linear system using the Rouché-Frobenius theorem and elementary transformations. The matrix A depends on the parameter m. Learn how to analyze this system step by step for different values of m.

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