Let S denote the set of all real values of such that the system of equations x + y + z = 1 , x + y + z = 1 , x + y + z = 1 is inconsistent. Then is equal to [2023]

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Solution

Q.
Let S denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1, x + λ y + z = 1, x + y + λ z = 1$ is inconsistent. Then $\sum_{λ \in S} (| λ |^{2} + | λ |)$ is equal to [2023]

1 12

2 4

3 2

4 6

Ans.
(4)

Since, the given system of equations is inconsistent

$\Rightarrow$ $| \begin{matrix} λ & 1 & 1 \\ 1 & λ & 1 \\ 1 & 1 & λ \end{matrix} |$ $= 0$ $\Rightarrow λ (λ^{2} - 1) - 1 (λ - 1) + 1 (1 - λ) = 0$

$\Rightarrow λ^{3} - λ - λ + 1 + 1 - λ = 0 \Rightarrow (λ - 1) (λ^{2} + λ - 2) = 0$

$\Rightarrow λ = 1, λ = \frac{- 1 \pm \sqrt{1 + 8}}{2} \Rightarrow λ = 1, λ = \frac{- 1 \pm 3}{2}$

$\Rightarrow λ = 1, 1, - 2$

Now, for $λ = 1$ , we have infinite solutions as we have only one equation $x + y + z = 1$

For $λ = - 2$ , $D_{1} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix} |$ $= 9 \neq 0$
$\Rightarrow for λ = - 2, we have no solution.$

$\sum_{λ \in S} (| λ^{2} | + | λ |) = (4 + 2) = 6$

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