Question

Solve for $\lambda$ if $\left|\begin{array}{ccc}{a}^2+\lambda & ab& ac\\ ab& b^2+\lambda & bc\\ ac& bc& c^2+\lambda \end{array}\right|=0$.

Answer 1

$\left|\begin{array}{ccc}{a}^2+\lambda & ab& ac\\ ab& b^2+\lambda & bc\\ ac& bc& c^2+\lambda \end{array}\right|=0$

$\therefore (a^2+\lambda )\left(\right(b^2+\lambda \left)\right(c^2+\lambda )-b^2{c}^2)-ab\left(ab\right(c^2+\lambda )-ab{c}^2)+ac(a{b}^{2}c-ac(b^2+\lambda \left)\right)=0$

$\therefore (a^2+\lambda )(b^2{c}^2+b^2\lambda +c^2\lambda +{\lambda }^2-b^2{c}^2)-ab(ab{c}^2+ab\lambda -ab{c}^2)+ac(a{b}^{2}c-a{b}^{2}c-ac\lambda ))=0$

$\therefore (a^2+\lambda )(b^2\lambda +c^2\lambda +{\lambda }^2)-a^2{b}^2\lambda -a^2{c}^2\lambda =0$

$\therefore a^2(b^2\lambda +c^2\lambda +{\lambda }^2)+\lambda (b^2\lambda +c^2\lambda +{\lambda }^2)-a^2(b^2\lambda +c^2\lambda )=0$

$\therefore a^2\left({\lambda }^2\right)+\lambda (b^2\lambda +c^2\lambda +{\lambda }^2)=0$

$\therefore \lambda (a^2\lambda +b^2\lambda +c^2\lambda +{\lambda }^2)=0$

$\therefore {\lambda }^2(a^2+b^2+c^2+\lambda )=0$

$\therefore {\lambda }^2=0\text{or}{a}^2+b^2+c^2+\lambda =0$

$\therefore \lambda =0\text{or}\lambda =-(a^2+b^2+c^2)$

These are the required values of $\lambda$.