Matrices & Determinants
System of linear equations — Cramer's rule
Grade Class 12
Question:
The system of linear equations x + λy - z = 0, λx - y - z = 0, x + y - λz = 0 has a non-trivial solution for :
(1) exactly three values of λ.
(2) infinitely many values of λ.
(3) exactly one value of λ.
(4) exactly two values of λ.
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 system is homogeneous. For a non-trivial solution, the determinant of the coefficient matrix must be zero: |1 \lambda -1; \lambda -1 -1; 1 1 -\lambda| = 0. Expanding the determinant: 1(\lambda - (-1)) - \lambda(-\lambda^2 - (-1)) - 1(\lambda - (-1)) = 0. Simplifying gives: (\lambda+1) - \lambda(1-\lambda^2) - (\lambda+1) = 0, which simplifies to \lambda^3 - \lambda = 0. Thus, \lambda(\lambda-1)(\lambda+1) = 0. The values are \lambda = 0, 1, -1. There are exactly three values.
Correct Answer: 1