Mathematics - Determinant Question with Solution | TestHub

$|\mathrm{A}|=0$ as the matrix $\mathrm{A}$ is singular $$\therefore |\mathrm{A}|=\left|\begin{array}{ccc}1& 3& \lambda +2\\ 2& 4& 8\\ 3& 5& 10\end{array}\right|=0$$ Apply ${\mathrm{R}}_2\to {\mathrm{R}}_2-2{\mathrm{R}}_1$ and ${\mathrm{R}}_3\to {\mathrm{R}}_3-3{\mathrm{R}}_1$ and expand. $-2(4-3\lambda )+4(4-2\lambda )=0$ $$\Rightarrow 8-2\lambda =0\Rightarrow \lambda =4$$ For $\lambda =4,$ the second and the third column are proportional.