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Let’s go through each part of this assignment step-by-step.

### 1. Solve the System Using Gaussian Elimination
The first task involves solving a system of equations through Gaussian elimination:
\[
\begin{cases}
x + y + z = 0 \\
y - z = 0 \\
x + 2y = 0
\end{cases}
\]
1. Write this system as an augmented matrix and use row operations to bring it to reduced row echelon form.
2. Identify if the system has a unique solution, infinite solutions, or no solution, and find the solutions.

### 2. Find the Inverse of Matrix \( A \) and Verify
The matrix \( A \) is given as:
\[
A = \begin{bmatrix} 1 & 2 & 4 \\ 2 & 3 & 4 \\ 1 & 0 & 1 \end{bmatrix}
\]
1. Use methods like the adjoint method, row reduction, or matrix formulas to find \( A^{-1} \).
2. Verify the inverse by showing that \( A \times A^{-1} = I \) (the identity matrix).

### 3. Solve the System Using the Inverse of a Matrix
The final task is to solve another system of equations by using the inverse matrix method:
\[
\begin{cases}
5x + 15y + 56z = 35 \\
4x + 11y + 41z = 26 \\
x + 3y + 11z = 7
\end{cases}
\]
1. Formulate this system in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \( B \) is the constants matrix.
2. Solve for \( X \) by calculating \( X = A^{-1} B \).

Would you like the detailed solutions for each step? Let me know if you’d like me to start with any specific part.

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Here are some related questions that expand upon these topics:

1. How do you determine if a system of linear equations has unique, infinite, or no solutions?
2. What are the main differences between Gaussian elimination and Gauss-Jordan elimination?
3. How can you verify that a calculated inverse matrix is correct?
4. Why is the inverse matrix method useful for solving systems of linear equations?
5. What conditions must a matrix satisfy to have an inverse?

**Tip:** Always check if a matrix is invertible by confirming that its determinant is non-zero.

Find the solution of the given system using Gaussian elimination, find the inverse of a matrix and check the result, and solve another system using the inverse of a matrix.

Explore methods for solving systems of linear equations using Gaussian elimination and matrix inversion. This includes finding the inverse of a matrix, verifying solutions, and applying matrix methods to solve linear equations. Ideal for undergraduate students studying linear algebra.

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