Consider a 2 2 system x = Ax. If we assume that r1 = r2, the general solution isx = c1

Chapter 7, Problem 28

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Consider a 2 2 system x = Ax. If we assume that r1 = r2, the general solution isx = c1 (1)er1t + c2 (2)er2t, provided that (1) and (2) are linearly independent. In this problemwe establish the linear independence of (1) and (2) by assuming that they are linearlydependent and then showing that this leads to a contradiction.(a) Note that (1) satisfies the matrix equation (A r1I) (1) = 0; similarly, note that(A r2I) (2) = 0.(b) Show that (A r2I) (1) = (r1 r2) (1).(c) Suppose that (1) and (2) are linearly dependent. Then c1 (1) + c2 (2) = 0 and at leastone of c1 and c2 (say c1) is not zero. Show that (A r2I)(c1 (1) + c2 (2)) = 0, and also showthat (A r2I)(c1 (1) + c2 (2)) = c1(r1 r2) (1). Hence c1 = 0, which is a contradiction.Therefore, (1) and (2) are linearly independent.(d) Modify the argument of part (c) if we assume that c2 = 0.(e) Carry out a similar argument for the case in which the order n is equal to 3; note thatthe procedure can be extended to an arbitrary value of n.