This is a homework question for a first course in real analysis (tiny Rudin) so I'd appreciate hints whilst straight out answers are discouraged due to academic honesty.
I'm given recursively defined: $M_0=I$, $M_k=I+\int_{t_0}^tA(s)M_{k-1}(s)ds$. Where A is continuous on $[t_0,t_1]$ I've (with a bit of help) shown for matrix M, A (n x n) that this converges uniformly to some function $\Phi(t)$ that satisfies $\Phi' =A\Phi, \,\Phi(t_0)=I$.
Now I need to show that for $det(\Phi)=exp(\int_{t_0}^ttr(A(s)ds)$ and that for scalar continuous $A$, $\Phi=exp(\int_{t_0}^t(A(s)ds)$.
Ive worked a bit on the exponential representation and realized that the recursive definition wasn't much help. However, the differential equation for $\Phi$ reminds very much of one of the definitions of the exponential functions. On the determinant part I hardly know where to begin.
EDIT: Ive just realized contraction mapping could be useful
I'd appreciate any pointers or links to reference literature (I did look at a couple of DE books but they didnt really help).
Thanks in advance.