Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is
Step-by-Step Solution
Key Concept: For a homogeneous system of linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero. The determinant is a circulant determinant involving roots of the cubic equation.
The system has a non-trivial solution if the determinant of the coefficient matrix is zero: |\alpha \beta \gamma; \beta \gamma \alpha; \gamma \alpha \beta| = 0. This simplifies to -(\alpha+\beta+\gamma)(\alpha^2+\beta^2+\gamma^2-\alpha\beta-\beta\gamma-\gamma\alpha) = 0. Since \alpha, \beta, \gamma are roots of x^3+ax^2+bx+c=0, we have \alpha+\beta+\gamma = -a and \alpha\beta+\beta\gamma+\gamma\alpha = b. The condition becomes -(\alpha+\beta+\gamma)((\alpha+\beta+\gamma)^2-3(\alpha\beta+\beta\gamma+\gamma\alpha)) = 0, which is -(-a)(a^2-3b) = 0. Since a \neq 0, we have a^2-3b = 0, so a^2/b = 3.
Correct Answer: 2