Question

Find tbe sum of all the values of $\lambda$ for which the system of equations $\lambda x+y+z=1$, $x+\lambda y+z=\lambda$, $x+y+\lambda z={\lambda }^2$ has infinite number of solutions.

Answer 1

$letAbethecoefficientmatrix,\left|A\right|=\left|\begin{array}{ccc}\lambda & 1& 1\\ 1& \lambda & 1\\ 1& 1& \lambda \end{array}\right|={(\lambda -1)}^2(\lambda +2)if\Delta =0,{\Delta }_1,{\Delta }_2,{\Delta }_3\ne 0inconsistent\Delta =0,anyof{\Delta }_1,{\Delta }_2,{\Delta }_{3}iszeroinfinitesolutionsfor\lambda =1\Delta =0,all{\Delta }_1,{\Delta }_2,{\Delta }_3=0\lambda =-2\Delta =0,{\Delta }_1,{\Delta }_2,{\Delta }_3\ne 0Hencefor\lambda =1givensystemhasinfinitenoofsolutions.$