Let S denote the set of all real values of [] such that the system of equations [ x+y+z=1]…

Question

Let $S$ denote the set of all real values of $$\lambda$$ such that the system of equations

$$\lambda x+y+z=1$$

$$x+\lambda y+z=1$$

$$x+y+\lambda z=1$$

is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$$ is equal to

Accepted Answer

$\left|\begin{array}{lll} \lambda & 1 & 1 \ 1 & \lambda & 1 \ 1 & 1 & \lambda \end{array}\right|=0$

$$
\begin{aligned}
& \lambda\left(\lambda^{2}-1\right)-1(\lambda-1)+1(1-\lambda)=0 \\
& \Rightarrow \lambda^{3}-\lambda-\lambda+1+1-\lambda=0 \\
& \Rightarrow \lambda^{3}-3 \lambda+2=0 \\
& \Rightarrow (\lambda-1)\left(\lambda^{2}+\lambda-2\right)=0
\end{aligned}
$$

$\Rightarrow$ $\lambda=1,-2$

For $\lambda=1 \Rightarrow \infty$ solution

$\lambda=-2 \Rightarrow$ no solution

$\sum\limits_{\lambda \in S}|\lambda|^{2}+|\lambda|=6$

Solutions · 1