[help] A matrix problem??

[v9260019]

hello----all
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

 

[steve10]

The result by "method 3" is right and all others are incorrect.
This can be done manually. Note that
A^n={{1,0,-n},{0,1,0},{0,0,1}}
Therefore,
Exp[At]=I+At/1!+(At)^2/2! +...
The only thing special is the element at the upper right corner which is

-t/1!-2t^2/2! - 3t^3/3! - ...
=-t(1+t/1! +t^2/2! +...)=-tExp[t].