We can copy Taylor's series for ex to define eAt for a matrix A. If A is diagonalizable, we can use Λ to find the exact value of eAt. This allows us to solve systems of differential equations du / dt = Au the same way we solved equations like dy / dt = ky.
Differential Equations and exp(At)
Differential Equations exp(At)
Work the problems on your own and check your answers when you're done.