Question

Find the value of $\lambda$ for which the given system has non-trivial solution.
$7x+4y+3z=0$
$x+2y+\lambda z=0$
$x+3y+2z=0$.

Answer 1

$\begin{array}{c}\left(\begin{array}{ccc}7& 4& 3\\ 1& 2& \lambda \\ 1& 3& 2\end{array}\right)=0\\ \Rightarrow 7\left(4-3\lambda \right)-4\left(2-\lambda \right)+\left(3-2\right)=0\\ \Rightarrow 28-21\lambda -8+4\lambda +3=0,\\ -17\lambda +23=0\\ \therefore \lambda =\frac{23}{17}\end{array}$