1. In this problem A is the 3 x 3 matrix 41 -4 A = 03-2 11-1 i. Show that the eigenvalues of A are the numbers 1, 2 and 3. ii. Find eigenvectors v(1), v(2) and v(3) corresponding to these three eigenvalues. iii. If you consider A to be the matrix of a linear transformation L from RS to RS relative to the usual basis, what is the matrix of L relative to the basis v(1), v(2), v(3). (That is, express L of each v as a linear combination of the v's.) Call this matrix B. iii. Use the results of i and ii to solve the system of simultaneous linear differential equations X' = AX, where f, g and h are differentiable functions and f X = 8 h iv. Find the solution to the differential system with f(0)= g(0) = h(0) = 1. V. Compute B" and eBt . vi. If M is the 3 x 3 matrix that gives the vectors v(1), v(2) and v(3) as columns in turns of the usual basis, find the inverse of M- call it N. vii. Use v and vi, or some other method, to find the powers of A and eAt. viii. Solve the system X' = AX + C, where C is the column vector