I've attempted to solve this by separating A into a diagonal matrix D and nilpotent matrix N:
D = {{1, 0}, {0, 0}}
N = {{0, 0}, {1, 0}}
e^(At) = e^((D + N)t) = e^(Dt) * e^(Nt)
When N is raised to the second power, it becomes the zero matrix. Therefore,
e^(Nt) = I + Nt = {{1, 0}, {t, 1}}
Note...