Let (t) denote the fundamental matrix satisfying = A,(0) = I. In the text we alsodenoted

Chapter 7, Problem 15

(choose chapter or problem)

Let (t) denote the fundamental matrix satisfying = A,(0) = I. In the text we alsodenoted this matrix by exp(At). In this problem we show that does indeed have theprincipal algebraic properties associated with the exponential function.(a) Show that (t)(s) = (t + s); that is, show that exp(At) exp(As) = exp[A(t + s)].Hint: Show that if s is fixed and t is variable, then both (t)(s) and (t + s) satisfy theinitial value problem Z = AZ, Z(0) = (s).(b) Show that (t)(t) = I; that is, exp(At) exp[A(t)] = I. Then show that(t) = 1(t).(c) Show that (t s) = (t)1(s)