If the system of equations 

 $\lambda x_1+x_2+x_3=1$ , $x_1+\lambda x_2+x_3=1$ , $x_1+x_2+\lambda x_3=1$
 is inconsistent, then  $\lambda$ equals 
 

Let  $\Delta =\left|\begin{array}{ccc}\lambda & 1& 1\\ 1& \lambda & 1\\ 1& 1& \lambda \end{array}\right|=\left|\begin{array}{ccc}\lambda +2& 1& 1\\ \lambda +2& \lambda & 1\\ \lambda +2& 1& \lambda \end{array}\right|\left[C_1\to C_1+C_2+C_3\right]$ 

$\begin{array}{l}=\left(\lambda +2\right)\left|\begin{array}{ccc}1& 1& 1\\ 1& \lambda & 1\\ 1& 1& \lambda \end{array}=\right|\left|\begin{array}{ccc}1& 0& 0\\ 1& \lambda -1& 0\\ 1& 0& \lambda -1\end{array}\right|\left[{\text{using C}}_2\to C_2-C_1{\text{and\hspace{0.17em} C}}_3\to C_2-C_1\right]\\ =\left(\lambda +2\right){\left(\lambda -1\right)}^2\end{array}$
If  $\Delta =0,\text{then }\lambda =-2\text{\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}\lambda =1$
But when  $\lambda$=1, the system of equation becomes  $x_1+x_2+x_3=1$ which has infinite number of solutions. When$\lambda =-2$ , by adding three equations, we obtain $0=3$  and thus, the system of equations is inconsistent.