Question

Let $S$ denote the sum of all the values of $\lambda$ for which the system of equations
$(1+\lambda )x_1+x_2+x_3=1$
$x_1+(1+\lambda )x_2+x_3=\lambda$
$x_1+x_2+(1+\lambda )x_3={\lambda }^2$
is inconsistent. Find $|S|$.

Answer 1

$\left|\begin{array}{ccc}1+\lambda & 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right|$

$\Rightarrow (1+\lambda )({\lambda }^2+2\lambda )-1\left(\lambda \right)+1(-\lambda )=0$

$\Rightarrow {\lambda }^2+2\lambda +{\lambda }^3+2{\lambda }^2-2\lambda =0$

$\Rightarrow {\lambda }^3+3{\lambda }^2=0$

$\Rightarrow {\lambda }^2(\lambda +3)=0$

$\Rightarrow \lambda =0,0,-3$

$S=0+0-3=-3$

$\left|S\right|=3$

Correct answer is $3$