Let $S$ denote the set of all values of $t$ such that the system of homogeneous equations
$t x + (t + 1) y + (t - 1) z = 0$
$(t + 1) x + t y + (t + 2) z = 0$
$(t - 1) x + (t + 2) y + t z = 0$
has non-trivial. Find
$4 \sum t \in S (- t)$

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Solution

$S y s t e m o f e q u a t i o n s h a s a n o n - t r i v i a l s o l u t i o n i . e Δ = 0 l e t A b e t h e c o e f f i c i e n t m a t r i x, | A | = ∣ ∣ ∣ ∣ \begin{matrix} t & t + 1 & t - 1 t + 1 & t & t + 2 t - 1 & t + 2 & t \end{matrix} ∣ ∣ ∣ ∣ = - 4 (2 t + 1) = 0 \Rightarrow t = \frac{- 1}{2} H e n c e 4 \sum (- t) = 2$

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