We are given a system of two linear equations:

$x_1 + 2x_1 + 3x_3 + 4x_4 = 0$ $2x_1 + 4x_2 + \lambda_1 x_3 + \lambda_2 x_4 = 0$

This simplifies to:

1. $3x_1 + 3x_3 + 4x_4 = 0$
2. $2x_1 + 4x_2 + \lambda_1 x_3 + \lambda_2 x_4 = 0$

We are interested in determining for what values of the parameters $\lambda_1$ and $\lambda_2$ the system has nonzero solutions, i.e., the system is not trivially solvable.

Rewrite as Matrix Form

The system can be written in matrix form as:

3 & 0 & 3 & 4 \\ 2 & 4 & \lambda_1 & \lambda_2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$ For the system to have non-trivial (nonzero) solutions, the matrix must be **singular**, meaning its determinant should be zero. ### Compute the Determinant The determinant of the $$2 \times 4$$ matrix is based on the square matrix formed by selecting columns (since a determinant doesn't exist directly for a non-square matrix). The matrix to consider is: $$\text{det}\begin{pmatrix} 3 & 3 & 4 \\ 2 & \lambda_1 & \lambda_2 \end{pmatrix}$$ Expanding the determinant: $$\text{det} = 3(\lambda_1 \lambda_2 - 0) - 3(2\lambda_2 - 0) + 4(2\lambda_1 - 0)$$ $$\text{det} = 3 \lambda_1 \lambda_2 - 6\lambda_2 + 8\lambda_1$$ ### Condition for Non-trivial Solutions For the system to have nonzero solutions, we need the determinant to be zero: $$3 \lambda_1 \lambda_2 - 6\lambda_2 + 8\lambda_1 = 0$$ ### Factor the Equation We can factor the equation to simplify it: $$\lambda_2(3 \lambda_1 - 6) + 8 \lambda_1 = 0$$ $$\lambda_2(3 \lambda_1 - 6) = -8 \lambda_1$$ ### Analyze the Range of $$\lambda_1$$ and $$\lambda_2$$ The next step is to determine when this equation is satisfied. Solving explicitly for $$\lambda_1$$ and $$\lambda_2$$ gives conditions on the parameters where nontrivial solutions exist. Would you like a detailed step-by-step solution for solving this factorization? Also, if this problem raises any specific questions, feel free to ask! Here are some related questions: 1. What happens if $$\lambda_1 = 0$$? 2. How do we interpret solutions geometrically for this system? 3. How can we extend this analysis to higher-dimensional systems? 4. What methods can be used to check if a matrix is singular? 5. How does this system change if the right-hand side isn't zero? **Tip**: Singular matrices are key to finding non-trivial solutions in homogeneous systems.