Let A = " −1 1 1 −1 # . (a) Find the eigenvalues and eigenvectors, and the exponential e At . (b) Compute d dt(e At) by differentiating each component and confirm that d dt(e At) = AeAt . (c) Solve the system of the differential equations du1(t) dt = −u1(t) + u2(t), du2(t) dt = u1(t) − u2(t), with the initial condition u1(0) = 2, u2(0) = 5.

1. an economy consists of three workers: larry, moe, and curly. each works 10 hours a day and can produce two services: mowing lawns and washing cars. in an hour, larry can either mow one lawn or wash one car; moe can either mow one lawn or wash two cars; and curly can either mow two lawns or wash one car. a. calculate how much of each service is produced under the following circumstances: a. all three spend all their time mowing lawns. b. all three spend all their time washing cars. c. all three spend half their time on each activity d. larry spends half of his time on each activity, while moe only washes cars and curly only mows lawns b. using your answers to part a, draw a production possibilities frontier and label points a, b, c, and d on your graph. explain why the production possibilities frontier has the shape it does. c. are any of the allocations calculated in part a inefficient? explain.