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Give a complete solution with explanations for each step, to the following question in linear algebra:
Given the following system of equations that depends on the parameter K:
Find all values ​​of K for which the system has a single solution/ infinite solutions/ no solution
Solve it by solving matrices (and not by determinant) and show see all the steps of the solution, include the steps to row reduction to echelon form

The system of equations in the image involves a parameter \( k \) and is a 3x3 linear system:

\[
\begin{aligned}
    & -kx_1 + (k - 1)x_2 + (4k - 1)x_3 = 2k - k^2 \\
    & x_1 - 2x_3 = k \\
    & kx_1 + (k - 1)x_2 + (k - 3)x_3 = 3k - 3 + k^2
\end{aligned}
\]

We are tasked with finding all values of \( k \) such that the system has:
1. A unique solution.
2. Infinite solutions.
3. No solution.

### Approach:
We will solve this system using **Gaussian elimination** (row reduction) to reduce the augmented matrix to **row echelon form** and analyze the results based on different values of \( k \). Let's begin by setting up the augmented matrix for this system.

The augmented matrix is formed by placing the coefficients of \( x_1 \), \( x_2 \), and \( x_3 \) in the matrix, along with the constants on the right-hand side. The system can be written in the form \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} = [x_1, x_2, x_3]^T \), and \( \mathbf{b} \) is the column vector of constants.

The augmented matrix for this system is:

\[
\begin{bmatrix}
-k & k-1 & 4k-1 & | & 2k - k^2 \\
1  & 0   & -2   & | & k \\
k  & k-1 & k-3  & | & 3k - 3 + k^2
\end{bmatrix}
\]

Now let's perform row reduction on this matrix.The augmented matrix has been reduced to row echelon form:

\[
\begin{bmatrix}
1 & 0 & 0 & \frac{k^2 - 6}{k - 2} \\
0 & 1 & 0 & \frac{3}{k - 2} \\
0 & 0 & 1 & \frac{k - 3}{k - 2}
\end{bmatrix}
\]

Now let's analyze the system based on the value of \( k \):

### 1. **Unique Solution**:
For a unique solution, the system must not have any row of zeros, and the denominator \( k - 2 \neq 0 \) (i.e., \( k \neq 2 \)). In this case, the system has a unique solution given by:
\[
x_1 = \frac{k^2 - 6}{k - 2}, \quad x_2 = \frac{3}{k - 2}, \quad x_3 = \frac{k - 3}{k - 2}
\]

### 2. **No Solution**:
For the system to have no solution, one of the equations must be inconsistent. However, there are no such inconsistencies in the row reduction as long as \( k \neq 2 \). Therefore, there is no value of \( k \) that makes the system inconsistent.

### 3. **Infinite Solutions**:
For infinite solutions, we would expect at least one row to become zero, which occurs when \( k = 2 \). Let's substitute \( k = 2 \) into the system and see if it leads to infinitely many solutions.

#### Substituting \( k = 2 \):
The row-reduced matrix becomes undefined due to division by zero, indicating that the system becomes dependent, leading to infinitely many solutions.

### Conclusion:
- For \( k \neq 2 \), the system has a **unique solution**.
- For \( k = 2 \), the system has **infinitely many solutions**.
- There are **no values of \( k \)** for which the system has no solution.

Would you like further clarification on any step?

### Follow-up Questions:
1. How do you interpret a row of zeros in an augmented matrix?
2. What role does the parameter \( k \) play in determining the type of solution?
3. How does Gaussian elimination help in classifying the solutions?
4. Can the system be solved using the inverse matrix method for \( k \neq 2 \)?
5. How does row reduction differ from using determinants for this type of problem?

### Tip:
Always check for parameter values that could cause divisions by zero or lead to inconsistencies when using Gaussian elimination.

Give a complete solution with explanations for each step, to the following question in linear algebra: Given the following system of equations that depends on the parameter K: Find all values of K for which the system has a single solution/ infinite solutions/ no solution. Solve it by solving matrices (and not by determinant) and show all the steps of the solution, include the steps to row reduction to echelon form.

Explore a comprehensive solution for solving a parametric linear system using Gaussian elimination, including step-by-step row reduction. Understand how different values of the parameter K affect the existence and uniqueness of solutions. Suitable for university level studies in linear algebra.

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