1 0 -1 a) Let A = 0 1 1 . Find A-l. 2 0. -3 3 -1 0 b) Given that B = -1 2 -1 O -1 3 i) Find the eigenvalues of B. ii) Given that the eigenvectors for matrix B are corresponding to eigenvalues 1, 3 and 4 respectively. Find the matrix P and the diagonal matrix D such that matrix B can be diagonalized via P-'BP = D. iii) Given that the differential equations modelling a certain control system are x = 3x1 -X2 x, = -x +2x2 -*3 X, =-*2 +3x, Using the solutions from (ii) above, find the general solution for x, , x, and.9 ._ 4 a) Suppose that a matrix A is given as follows: ,4 = [8 3]. i) Show that the eigenvalues ofA are I and 5 by formulating the characteristic equation. ii) For one of the eigenvalue, nd its eigenvector. iii) Find the eigenvalues of A". b) A system of differential equation is given as follows: I. I an 9x: ~4x2 x; = Sal 3:r2 9 4 where x, and x2 are functions oft. Given that. the eigenvalues of A =l:8 J are 1 l l and 5 with the corresponding eigenvectors [2] and [I , respectively. i) Write a matrix P that diagonalizes A. ii) Write a diagonal matrix D such that P'IAP = D. iii) Hence, find the general solution to the system