Question

For which rational numbers does the homogenous system
x+(λ-3)y=0
(λ-3)x+y=0
Have a non-trivial solution

Answer 1

$\text{The co-efficient matrix for the given system is} \begin{pmatrix} 1 & \lambda - 3 \\ \lambda -3 & 1 \end{pmatrix}\\ \text{For given system of linear equations to have a non-trivial solution, then its determinant}\\ \text{be 0.}\\ \begin{vmatrix} 1 & \lambda - 3 \\ \lambda -3 & 1 \end{vmatrix}= 1-(\lambda -3)^2 =0\\ = -\lambda^2 + 6\lambda-8=0\\ \implies \text{$\lambda = -2$ or $\lambda=-4$}$