Let the system of linear equations 4 x+λ y+2 z=0 2 x-y+z=0 μ x +2 y +3 z =0, λ, μ ∈ R has a non-trivial solution. Then which of the following is true?

Question

Let the system of linear equations 4 x+λ y+2 z=0 2 x-y+z=0 μ x +2 y +3 z =0, λ, μ ∈ R has a non-trivial solution. Then which of the following is true? (A) μ=6, λ ∈ R (B) λ=2, μ ∈ R (C) λ=3, μ ∈ R (D) μ=-6, λ ∈ R

Tardigrade Admin · Accepted Answer

For non-trivial solution |4 λ 2 2 -1 1 μ 2 3|=0 ⇒ 2 μ-6 λ+λ μ=12 when μ=6, 12-6 λ+6 λ=12 which is satisfied by all λ

Q. Let the system of linear equations
$4 x + λ y + 2 z = 0$
$2 x - y + z = 0$
$μx + 2 y + 3 z = 0, λ, μ \in R$
has a non-trivial solution. Then which of the following is true?

$μ = 6, λ \in R$

$λ = 2, μ \in R$

$λ = 3, μ \in R$

$μ = - 6, λ \in R$

For non-trivial solution
$∣ ∣ 42 μ λ - 1 2 213 ∣ ∣ = 0$
$\Rightarrow 2 μ - 6 λ + λ μ = 12$
when $μ = 6, 12 - 6 λ + 6 λ = 12$
which is satisfied by all $λ$

Q. Let the system of linear equations 4x+λy+2z=0 2x−y+z=0 μx+2y+3z=0,λ,μ∈R has a non-trivial solution. Then which of the following is true?

μ=6,λ∈R

λ=2,μ∈R

λ=3,μ∈R

μ=−6,λ∈R