First slide

System of linear equations

Question

The set of all values of $λ$ for which the system of linear equations $2 x_{1} - 2 x_{2} + x_{3} = λ x_{1}$ , $2 x_{1} - 3 x_{2} + 2 x_{3} = λ x_{2}$ , $- x_{1} + 2 x_{3} = λ x_{3}$ has a non-trivial solution

Moderate

A

Is an empty set
B

Is a single ton
C

contains two elements
D

contains more than two elements

Solution

$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m o f e q u a t i o n s t o h a v e a n o n - t r i v i a l s o l u t i o n i s Δ = 0$
$\Rightarrow$ $|\begin{array}{l} 2 - λ - 2 1 \\ 2 - 3 - λ 2 \\ - 1 2 - λ \end{array}| = 0$
$R_{1} \to R_{1} + R_{3}, R_{2} \to R_{1} + 2 R_{3}$
$\Rightarrow |\begin{array}{l} 1 - λ 0 1 - λ \\ 0 1 - λ 2 - 2 λ \\ - 1 2 - λ \end{array}| = 0 \Rightarrow {(1 - λ)}^{2} |\begin{array}{l} 1 0 1 \\ 0 1 2 \\ - 1 2 - λ \end{array}| = 0$
$C_{3} \to C_{3} + C_{1}$
$\Rightarrow {(1 - λ)}^{2} |\begin{array}{l} 1 0 0 \\ 0 1 2 \\ - 1 2 - λ + 1 \end{array}| = 0$
$\Rightarrow {(1 - λ)}^{2} [- λ + 1 - 4] = 0 \Rightarrow λ = 1, - 3$

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