System of linear equations

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$I f t h e s y s t e m o f e q u a t i o n s$ $x + 2 y + 3 z = λ x, 3 x + y + 2 z = λ y, 2 x + 3 y + z = λ z$ $h a s n o n - t r i v i a l s o l u t i o n t h e n$ $λ =______$

Moderate

Solution

$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m o f e q u a t i o n s t o h a v e a n o n - t r i v i a l s o l u t i o n i s Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{matrix} 1 - λ & 2 & 3 \\ 3 & 1 - λ & 2 \\ 2 & 3 & 1 - λ \end{matrix}| = 0 \\ \Rightarrow (1 - λ) [(1 - λ)^{2} - 6] - 2 [3 - 3 λ - 4] + 3 [9 - 2 + 2 λ] = 0 \\ \Rightarrow (1 - λ) [λ^{2} - 2 λ - 5] - 2 [- 3 λ - 1] + 3 [7 + 2 λ] = 0 \\ \Rightarrow - λ^{3} + 3 λ^{2} + 15 λ + 18 = 0 \Rightarrow (λ - 6) (λ^{2} + 3 λ + 3) = 0 \Rightarrow λ = 6 \end{array}$

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System of linear equations

Remember concepts with our Masterclasses.

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

The condition for a homogeneous system of equations to have a non-trivial solution is Δ=0⇒1-λ2331-λ2231-λ=0⇒(1-λ)(1-λ)2-6-2[3-3λ-4]+3[9-2+2λ]=0⇒(1-λ)λ2-2λ-5-2[-3λ-1]+3[7+2λ]=0⇒-λ3+3λ2+15λ+18=0⇒(λ-6)λ2+3λ+3=0⇒λ=6

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$I f t h e s y s t e m o f e q u a t i o n s$ $x + 2 y + 3 z = λ x, 3 x + y + 2 z = λ y, 2 x + 3 y + z = λ z$ $h a s n o n - t r i v i a l s o l u t i o n t h e n$ $λ =______$