5. (25 points) Suppose we have a closed system of particles north and south of a permeable barrier. Initially, there are no particles to the north and so particles to the south, and xo = The population vector after k + 1 minutes is given by Xk+1 = Axk, where A 1.6 .25 .75 (a) What do the .6 and .4 in the matrix tell us about movement of particles in this system? (b) Find the characteristic polynomial for A, and use it to deduce that A has eigenvectors 2 = 1 and 22=0.35 = 7/20 C) Find eigenvectors V1, V2 for A corresponding to 21, 22, respectively. (d) Suppose xo = 4000v1 +1000v2. Use the eigenvalues to determine lim xx, if it exists. (e) In a nearby lab, there is another system of particles which is just the same except that whenever particles migrate from north to south across the barrier, half of those north-to-south migrating particles leak out of the system. Write the matrix B which models the data for this system