find the value of λ
such that the system s of equation x+λy+3z=0 , 4x+3y+λz=0, 2x+y+2z=0 has non-trivial solution

To find the value of \( \lambda \) such that the system of equations has a non-trivial solution, we need to solve the following system:

\[
\begin{aligned}
x + \lambda y + 3z &= 0 \\
4x + 3y + \lambda z &= 0 \\
2x + y + 2z &= 0 
\end{aligned}
\]

A non-trivial solution exists if the determinant of the coefficient matrix is zero, i.e., the system is dependent.

The coefficient matrix is:
\[
A = \begin{pmatrix}
1 & \lambda & 3 \\
4 & 3 & \lambda \\
2 & 1 & 2
\end{pmatrix}
\]

To find \( \lambda \), we calculate the determinant of \( A \) and set it equal to zero.

\[
\det(A) = \begin{vmatrix}
1 & \lambda & 3 \\
4 & 3 & \lambda \\
2 & 1 & 2
\end{vmatrix}
\]

Now, we compute the determinant step by step:

\[
\det(A) = 1 \cdot \begin{vmatrix} 3 & \lambda \\ 1 & 2 \end{vmatrix} 
        - \lambda \cdot \begin{vmatrix} 4 & \lambda \\ 2 & 2 \end{vmatrix} 
        + 3 \cdot \begin{vmatrix} 4 & 3 \\ 2 & 1 \end{vmatrix}
\]

First, we compute each of the 2x2 determinants:
\[
\begin{vmatrix} 3 & \lambda \\ 1 & 2 \end{vmatrix} = 3(2) - \lambda(1) = 6 - \lambda
\]
\[
\begin{vmatrix} 4 & \lambda \\ 2 & 2 \end{vmatrix} = 4(2) - \lambda(2) = 8 - 2\lambda
\]
\[
\begin{vmatrix} 4 & 3 \\ 2 & 1 \end{vmatrix} = 4(1) - 3(2) = 4 - 6 = -2
\]

Now substitute back into the determinant expression:
\[
\det(A) = 1(6 - \lambda) - \lambda(8 - 2\lambda) + 3(-2)
\]
Simplify:
\[
\det(A) = 6 - \lambda - \lambda(8 - 2\lambda) - 6
\]
\[
\det(A) = -\lambda - \lambda(8 - 2\lambda)
\]
\[
\det(A) = -\lambda - 8\lambda + 2\lambda^2
\]
\[
\det(A) = 2\lambda^2 - 9\lambda
\]

For a non-trivial solution, we set \( \det(A) = 0 \):
\[
2\lambda^2 - 9\lambda = 0
\]

Factor the quadratic equation:
\[
\lambda(2\lambda - 9) = 0
\]

Thus, \( \lambda = 0 \) or \( \lambda = \frac{9}{2} \).

Therefore, the values of \( \lambda \) for which the system has a non-trivial solution are \( \lambda = 0 \) and \( \lambda = \frac{9}{2} \).

Would you like more details on this process or have any questions?

### Related Questions:
1. What is a non-trivial solution in a system of equations?
2. How do you calculate the determinant of a 3x3 matrix?
3. What are the conditions for a system of linear equations to have a unique solution?
4. How does the value of \( \lambda \) affect the solution of the system?
5. What methods can be used to solve a system of equations other than determinants?

### Tip:
Always verify the determinant computation carefully to ensure no sign errors or miscalculations!

find the value of λ such that the system of equations x+λy+3z=0, 4x+3y+λz=0, 2x+y+2z=0 has non-trivial solution

Learn how to find the value of λ for which the system of equations x+λy+3z=0, 4x+3y+λz=0, 2x+y+2z=0 has non-trivial solutions. This involves calculating the determinant of a 3x3 matrix and solving a quadratic equation. Suitable for advanced high school students or undergraduate learners.

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