$$\begin{aligned} & |A-\lambda I|=0 \\ & \left|\begin{array}{ccc} 2-\lambda & a & 0 \\ 1 & 3-\lambda & 1 \\ 0 & 5 & b-\lambda \end{array}\right|=0 \\ & (2-\lambda)[(3-\lambda)(b-\lambda)-5]-a[b-\lambda-0]+0=0 \\ & (2-\lambda)\left[3 b-3 \lambda-b \lambda+\lambda^2-5\right]-a b+a \lambda=0 \\ & \lambda^3-(b+5) \lambda^2+(1-a+5 b) \lambda+(10-6 b+a b)=0 \\ & A^3-(b+5) A^2+(1-a+5 b) A+(10-6 b+a b) I=0 \\ & \Rightarrow \mathrm{b}+5=4,1-a+5 b=1,10-6 b+a b=21 \\ & \Rightarrow a=-5, b=-1 \\ & \Rightarrow 2 a+3 b=-13 \end{aligned}$$