Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Let the system of linear equations <br> 4x + λy + 2z = 0 <br> 2x - y + z = 0 <br> μx + 2y + 3z = 0, λ, μ ∈ R <br> has a non-trivial solution. Then which of the following is true ?
(1) μ = 6, λ ∈ R
(2) λ = 2, μ ∈ R
(3) μ = 3, μ ∈ R
(4) μ = -6, λ ∈ R
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 has a non-trivial solution if the determinant of the coefficient matrix is zero: <br> |4 \lambda 2| <br> |2 -1 1| = 0 <br> |\mu 2 3| <br> Expanding along the first row: <br> 4(-3 - 2) - \lambda(6 - \mu) + 2(4 + \mu) = 0 <br> 4(-5) - 6\lambda + \lambda\mu + 8 + 2\mu = 0 <br> -20 - 6\lambda + \lambda\mu + 8 + 2\mu = 0 <br> -12 - 6\lambda + \lambda\mu + 2\mu = 0 <br> \mu(\lambda + 2) - 6(\lambda + 2) = 0 <br> (\mu - 6)(\lambda + 2) = 0 <br> This implies \mu = 6 or \lambda = -2. Looking at the options, \mu = 6 is a valid condition.
Correct Answer: 1