Question

The system of equations
$(1+\lambda )x_1+x_2+x_3=1$
$x_1+(1+\lambda )x_2+x_3=\lambda$
$x_1+x_2+(1+\lambda )x_3={\lambda }^2$

• A. is in consistent if $\lambda =0$ or $\lambda =-3$
• B. has a unique solution if $\lambda (\lambda +3)\ne 0$
• C. has infinite number of solution for $\lambda \ne -3$
• D. none of these

Answer 1

Answer: (A), (B)

has a unique solution if $\lambda (\lambda +3)\ne 0$
is in consistent if $\lambda =0$ or $\lambda =-3$

$\left|\begin{array}{ccc}1+\lambda & 1& 1\\ 1& 1+\lambda & 1\\ 1& 1& 1+\lambda \end{array}\right|=(1+\lambda )({(1+\lambda )}^2-1)-1(1+\lambda -1)+1(1-1-\lambda )={\lambda }^2(\lambda +3)$