Task 9 (Matrix Exponential I) Let T: (0, oc) → L(C") satisfy • T(0) = Idx; • T(t + s) = T(t)T(s) (t, s 2 0); •T€ C(0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e' A, where A = †(0). Hint: Show that v(t) := T(s) ds is invertible for small to and for those T(t) = V(to)(V(t +to) - V(t)).

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A: Differential operator

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A: See the detailed solution below.

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