Consider linear equations of the form  x1 + 2x1 + 3x3+ 4x4 = 0, 2x1 + 4x2 + lambda_1 x3 + lambda_2 x4 = 0.   What is the range of parameters (lambda_1, lambda_2)    for which   the equations have nonzero solutions?

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:

\[
\begin{pmatrix}
3 & 0 & 3 & 4 \\
2 & 4 & \lambda_1 & \lambda_2
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{pmatrix}
= 
\begin{pmatrix}
0 \\
0
\end{pmatrix}
\]

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.

Consider linear equations of the form x1 + 2x1 + 3x3 + 4x4 = 0, 2x1 + 4x2 + lambda_1 x3 + lambda_2 x4 = 0. What is the range of parameters (lambda_1, lambda_2) for which the equations have nonzero solutions?

Explore the conditions under which a system of linear equations involving parameters lambda_1 and lambda_2 has nonzero solutions. This problem covers key linear algebra concepts such as matrix determinants, singular matrices, and homogeneous systems. Suitable for undergraduate students studying linear algebra.

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