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The second subquestion asks to solve the given linear system using **Gauss-Seidel method** and perform two iterations, starting from an initial guess of \(\mathbf{x}^0 = (0, 0)\). Here’s the detailed approach for the Gauss-Seidel method:

### Step 1: Write the linear system

The system of equations is given as:

\[
\begin{bmatrix}
5 & 2 \\
-1 & 4
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
1 \\
-5
\end{bmatrix}
\]

This corresponds to the following system of linear equations:
\[
5x_1 + 2x_2 = 1 \quad \text{(1)}
\]
\[
-x_1 + 4x_2 = -5 \quad \text{(2)}
\]

### Step 2: Solve for \(x_1\) and \(x_2\)

Using the Gauss-Seidel method, we rearrange the equations to express \(x_1\) in terms of \(x_2\), and \(x_2\) in terms of \(x_1\):

- From equation (1), solve for \(x_1\):
  \[
  x_1 = \frac{1 - 2x_2}{5}
  \]

- From equation (2), solve for \(x_2\):
  \[
  x_2 = \frac{-5 + x_1}{4}
  \]

### Step 3: Iterative process

We start with an initial guess \(\mathbf{x}^0 = (0, 0)\) and perform two iterations.

#### Iteration 1:
- From the initial guess \(\mathbf{x}^0 = (0, 0)\), substitute \(x_2^0 = 0\) into the equation for \(x_1\):
  \[
  x_1^1 = \frac{1 - 2(0)}{5} = \frac{1}{5} = 0.2
  \]
  
- Now, use \(x_1^1 = 0.2\) to update \(x_2\):
  \[
  x_2^1 = \frac{-5 + 0.2}{4} = \frac{-4.8}{4} = -1.2
  \]

So, after the first iteration, we have:
\[
\mathbf{x}^1 = (0.2, -1.2)
\]

#### Iteration 2:
- Using \(x_2^1 = -1.2\), update \(x_1\):
  \[
  x_1^2 = \frac{1 - 2(-1.2)}{5} = \frac{1 + 2.4}{5} = \frac{3.4}{5} = 0.68
  \]

- Now, use \(x_1^2 = 0.68\) to update \(x_2\):
  \[
  x_2^2 = \frac{-5 + 0.68}{4} = \frac{-4.32}{4} = -1.08
  \]

So, after the second iteration, we have:
\[
\mathbf{x}^2 = (0.68, -1.08)
\]

### Final result after 2 iterations:
\[
\mathbf{x}^2 = (0.68, -1.08)
\]

Would you like further details on this method, or have any questions?

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Here are 5 related questions:
1. What is the difference between the Gauss-Seidel and Jacobi methods for solving linear systems?
2. How can you determine if the Gauss-Seidel method will converge for a given matrix?
3. What is the role of diagonal dominance in the convergence of iterative methods?
4. How does the number of iterations affect the accuracy of the Gauss-Seidel method?
5. How can over-relaxation techniques improve the convergence speed of iterative methods?

**Tip:** The convergence of the Gauss-Seidel method is usually faster if the matrix is diagonally dominant, meaning that the magnitude of each diagonal entry is larger than the sum of the magnitudes of the other entries in its row.

Solve the linear system using the Gauss-Seidel method and perform two iterations, starting from an initial guess of \(\mathbf{x}^0 = (0, 0)\).

This page demonstrates how to solve a 2x2 linear system using the Gauss-Seidel method. Starting from an initial guess of (0, 0), follow the step-by-step solution, including two iterations to find \(x_1\) and \(x_2\). Learn about the conditions for convergence and the diagonal dominance theorem.

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