Matrices & Determinants
Determinants
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<br>, λ, μ ∈ R<br>has a non-trivial solution. Then which of the following is true ? [JEE (Main) 2021]
(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>\lambda(\mu - 6) + 2(\mu - 6) = 0<br>(\lambda + 2)(\mu - 6) = 0<br>This implies \mu = 6 or \lambda = -2. Looking at the options, \mu = 6 is a valid condition.
Correct Answer: (1)