(a) Let A(t) be a nxn matrix with entries depending on a variablet. Alt) is such that dA (AP) =p A? dt dt for all integer p > 0. (42) (i) Assuming that d dA dA A+A dt dt dt prove that dA dA A A dt dt (ii) Express exp(A) in the form of a series. d da (iii) Calculate (exp(A)) in terms of A and dt dt