Question

Let $S$ denotes 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. Then |S| is

Answer 1

$\Delta =\left|\begin{array}{ccc}1+\lambda & 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right|$
By applying $R_1\to R_1+R_2+R_3,\Delta =(3+\lambda )\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right|$
By applying $C_1\to C_1-C_3,\Delta =(3+\lambda )\left|\begin{array}{ccc}0& 1& 1\\ 0& 1+\lambda & 1\\ -\lambda & 1& 1+\lambda \end{array}\right|\Rightarrow (3+\lambda )\left({\lambda }^2\right)=0\Rightarrow \lambda =-3,0$
For $\lambda =0,$ the given system of equations has infinite solutions.
$\therefore S=-3\Rightarrow |S|=3$