If the system of equations x+2y+3z=λx, 3x+y+2z=λy, 2x+3y+z=λz has non-trivial solution , then λ=……..

A
6
B
12
C
16
D
18

Solution:

Given system of equations ,

$(1 - λ) x + 2 y + 3 z = 0, 3 x + (1 - λ) + 2 z = 0, 2 x + 3 y + (1 - λ) z = 0$

$Has nontrivial solution \Rightarrow |\begin{array}{ccc} 1 - λ & 2 & 3 \\ 3 & 1 - λ & 2 \\ 2 & 3 & 1 - λ \end{array}| = 0$

$\begin{aligned} R_{1} + R_{2} + R_{3} \\ \Rightarrow |\begin{array}{ccc} 6 - λ & 6 - λ & 6 - λ \\ 3 & 1 - λ & 2 \\ 2 & 3 & 1 - λ \end{array}| \\ (6 - λ) |\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 1 - λ & 2 \\ 2 & 3 & 1 - λ \end{array}| \\ C_{2} - C_{1} \\ C_{3} - C_{1} \end{aligned}$

$\begin{aligned} \Rightarrow (6 - λ) |\begin{array}{cc} 1 & 0 & 0 \\ 3 & - λ - 2 & - 1 \\ 2 & 1 & - λ - 1 \end{array}| = 0 \\ \Rightarrow (6 - λ) [(λ + 1) (λ + 2) + 1] = 0 \\ \Rightarrow λ = 6, λ^{2} + 3 λ + 3 = 0 \end{aligned}$