Question

Find the number of values of $\lambda$ for which the homogeneous system of equations
$(a-\lambda )x+by+cz=0$
$bx+(c-\lambda )y+az=0$
$cx+ay+(b-\lambda )z=0$
has a non-trivial solution.

Answer 1

$Systemofequationshasanon-trivialsolutioni.e\Delta =0letAbethecoefficientmatrix,\left|A\right|=\left|\begin{array}{ccc}a-\lambda & b& c\\ b& c-\lambda & a\\ c& a& b-\lambda \end{array}\right|=0clearly,determinantvalueisacubicequationin\lambda Hencenoofvaluesof\lambda =3$