Mathematics - Determinant Question with Solution | TestHub

$\left(2-\lambda \right)x_1-2x_2+x_3=0$

$2x_1-\left(\lambda +3\right)x_2+2x_3=0$

$-x_1+2x_2-\lambda x_3=0$

The systems of linear equations will have a non-trivial solution

$\Rightarrow \left|\begin{array}{ccc}2-\lambda & -2& 1\\ 2& -\lambda -3& 2\\ -1& 2& -\lambda \end{array}\right|=0$

$\Rightarrow \left(2-\lambda \right)\left[{\lambda }^2+3\lambda -4\right]+2\left[-2\lambda +2\right]+1\left[4-\lambda -3\right]=0$

$2{\lambda }^2+6\lambda -8-{\lambda }^3-3{\lambda }^2+4\lambda -4\lambda +4+1-\lambda =0$

$-{\lambda }^3-{\lambda }^2+5\lambda -3=0$

${\lambda }^3+{\lambda }^2-5\lambda +3=0$

$\left(\lambda -1\right)\left({\lambda }^2+2\lambda -3\right)=0$

$\left(\lambda -1\right)\left(\lambda +3\right)\left(\lambda -1\right)=0$

$\Rightarrow \lambda =-3,1,1$