#### v9260019

##### Member level 2

i use three method to find Exp[At],where a is a matrix

A={{1,0,-1},{0,1,0},{0,0,1}}

Method 1: use computer (mathematica) find Exp[At]

-> Exp[At]={{Exp[t],1,Exp[-t]},{1,Exp[t],1},{1,1,Exp[t]}}

Method 2:use Cayley-Hamilton theorem

The answer is Exp[At]={{Exp[t],0,-Exp[t]t+Exp[t]t^2/2},{0,Exp[t],0},{0,0,Exp[t]}}

Method 3: use laplace transform

we all know the InverseLaplaceTransform[Inverse[sI-A]]=Exp[At]]

The answer is Exp[At]={{Exp[t],0,-Exp[t]t},{0,Exp[t],0},{0,0,Exp[t]}}

The three answers of Exp[At] are all different ,which one is right and what the relationship between the three methods ??????

Thanks a lot