Finding all the eigenvalues

Question

Questions:

Find all the eigenvalues

1. The solution of the vector-valued differential equation

                          dX/dt = AX(t)          with X(0) = [a1 ]
                                                                            [a2]
                                                                            [a3]

is given by: X(t) = Exp(tA)X(0) = e^tA(X(0))
Explain how you would compute e^tA, and then find the general solution when:

                                       A = [ 0   1    0]
                                             [ 0    0    1]
                                             [ -1   1    1]

Hint: First compute the characteristic polynomial and find the jordan canonical form. If B = S-1 AS how are solutions of dY/dt = B.Y(t) related to the solution X(t) in (1)?