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The problem asks to determine whether the given system of linear equations has a non-trivial solution. A non-trivial solution means that the solution is not the zero vector (i.e., not all variables are equal to zero).

The system of equations provided is:

1. \( 4x + 2y + z + 3w = 0 \)
2. \( 6x + 3y + 4z + 7w = 0 \)
3. \( 2x + y + w = 0 \)

To determine if a non-trivial solution exists, we can proceed by expressing this as a matrix equation \( A\mathbf{x} = 0 \), where \( A \) is the coefficient matrix and \( \mathbf{x} \) is the vector of unknowns \( [x, y, z, w]^T \).

Let's set up the augmented matrix for this system and find the solution through row reduction to identify if a non-trivial solution exists.

I'll calculate this now.The upper triangular matrix \( U \) from the LU decomposition is:

\[
U = \begin{pmatrix}
6 & 3 & 4 & 7 \\
0 & 0 & -1.67 & -1.67 \\
0 & 0 & -1.33 & -1.33
\end{pmatrix}
\]

Since there are two non-zero rows and one row of all zeros in the bottom row, the rank of the matrix \( A \) is 2, which is less than the number of variables (4). This indicates that the system has infinitely many solutions, including non-trivial solutions.

Thus, the system of equations does have a **non-trivial solution**.

Would you like a detailed step-by-step breakdown or further clarifications on how to solve it? 

Here are 5 related questions to expand your understanding:
1. How do you identify if a system has a unique solution, no solution, or infinitely many solutions?
2. What is the geometric interpretation of a system with a non-trivial solution?
3. Can LU decomposition always be used to find the rank of a matrix?
4. How can Gaussian elimination be applied to this system to find the solution?
5. What is the significance of the rank of a matrix in solving linear systems?

**Tip:** The rank of a matrix plays a crucial role in determining the type of solutions a system of linear equations can have (trivial, non-trivial, or inconsistent).

Explain whether the following equations have a non-trivial solution, and solve them:
1. 4x + 2y + z + 3w = 0
2. 6x + 3y + 4z + 7w = 0
3. 2x + y + w = 0

This problem involves determining if a system of linear equations has a non-trivial solution using matrix rank and LU decomposition. We explore the rank of the coefficient matrix to find if infinitely many solutions exist. Suitable for college-level students studying Linear Algebra.

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