I don't need the template because this isn't a homework problem, per se; it's just information about how to get started on my homework. So I have an equation x'=Ax, x(0)=x0, where A is a constant matrix. I can write it as x=ø(t)x0, where ø(t) is a fundamental matrix such that ø(0)=I We know that Taylor expanding eat gives us ∑ antn/n! (n starts from 1 and goes to infinity) I + ∑Antn / n! = I + At + A2t2/2! + ..... + Antn/n! + ........ = e(At) So, I don't even understand how you can raise a scalar to a power of a matrix. This is a messed-up world we live in. d/dt e(At) = ∑Antn-1/(n-1)! = A[ I + ∑Antn/n!]. I don't understand that last step. Must be some property of summations?