The image presents a system of linear equations $(S)$, expressed in matrix form as $AX = b$, where:

• $A = \begin{pmatrix} 5 & 2 & -1 \\ 1 & 6 & -3 \\ 2 & 1 & 4 \end{pmatrix}$,
• $X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$, and
• $b = \begin{pmatrix} 6 \\ 4 \\ 7 \end{pmatrix}$.

The tasks are as follows:

1. Show that there exists a unique solution to $(S)$ in $\mathbb{R}^3$.
2. Study the convergence of the Gauss-Seidel method for solving $(S)$.
3. Provide the iterative scheme for the Gauss-Seidel method associated with $(S)$.
4. Calculate the first three iterations using the Gauss-Seidel method.

Task 1: Show that there exists a unique solution of $(S)$ in $\mathbb{R}^3$

To prove the existence of a unique solution, we check if matrix $A$ is invertible, which can be confirmed by calculating its determinant:

$\text{det}(A) = \text{det}\begin{pmatrix} 5 & 2 & -1 \\ 1 & 6 & -3 \\ 2 & 1 & 4 \end{pmatrix}$

I'll calculate the determinant next to check if it's non-zero (a non-zero determinant indicates that $A$ is invertible, implying a unique solution).

Task 2: Study the convergence of the Gauss-Seidel method

The Gauss-Seidel method converges if the matrix $A$ is diagonally dominant, or if it satisfies certain other conditions related to its eigenvalues.

Matrix $A$ is diagonally dominant if for each row $i$, $|a_{ii}| > \sum_{j \neq i} |a_{ij}|$ I'll check whether matrix $A$ satisfies this property.

Task 3: Iterative scheme of the Gauss-Seidel method

For the Gauss-Seidel method, we solve for each variable sequentially from the system: