Math Problem Statement
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
Solution
To find the value of such that the system of equations has a non-trivial solution, we need to solve the following system:
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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Systems of Linear Equations
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Quadratic Equation: ax^2 + bx + c = 0
Theorems
Condition for non-trivial solutions: The determinant of the coefficient matrix must be zero for dependent systems.
Suitable Grade Level
Undergraduate level (Linear Algebra course) or advanced high school students