One way would be to use the fact that , where is defined as the sum of the series (of course, you have to show that the series converges in some suitable sense). Then the same proof which shows that can be used in this context.
Show that if Y (t); an n by n matrix, is the unique maximal solution of the
Y'(t) = A*Y(t); Y (0) = I;
then Y has the characteristic property of exponentials, namely
Y (s + t) = Y (s)Y (t) for all s; t E(element of) R.
Y (-t) = Y(t)^-1