Evaluate e^(tA) where A =[2,3,-3;
4,2,-4;
4,3,-5]

I have no idea how to start. If someone could explain the process that would be great.

Question

Evaluate e^(tA) where A =[2,3,-3;
                                         4,2,-4;
                                          4,3,-5]

I have no idea how to start. If someone could explain the process that would be great.

goformit100 · Answer

4,2,-4;

anonymous · Answer

What do you mean?

anonymous · Answer

step1 diagonalize A by finding out the eigenvalues eigenvectors. Using characteristic equation. I got eigenvalues are: 2, -2, and -1. Calculate the eigenvectors correspond to those eigenvalue. You will have P , find P^-1 . use formula $$\huge\P^-AP = D$$ 
to get D . That D is your matrix using to calculate e^At . the number in main diagonal line of D is the number of A in e^At
for example. If (I say If because I don't calculate ) your D is $$\left[\begin{matrix}2 &  0&0\ 0 & -2&0\0&0&-1\end{matrix}\right]$$. when plug it into the formula $$\huge\e^At = P A P^-$$ it will have the form something like 
$$e^At= P (\left[\begin{matrix}e^2t & 0&0 \ 0 & e^-2t&0\0&0&e^t\end{matrix}\right]P^-$$
you still have 1 more step is calculate that e^At
hope this helps. It's not easy or a piece of cake. It take long time to find out the answer. Fortunately, we have a path to do. You can do it, right? Good luck

anonymous · Answer

all e^(At), just typo. don't make mistake there. I mean don't think (e^A)* t