If the matrix [ array ccc 1 & 3 & λ+2 2 & 4 & 8 3 & 5 & 10 array ]...

$|\mathrm{A}|=0$ 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} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}$and$\mathrm{R}_{3}$\rightarrow$\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.$