Turn over 23 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 15 marks) 24 8. A = 5 2 5 0 3 6 6 4 − − −         p       where p is a constant Given that 2 1 2 −         is an eigenvector for A (a) (i) determine the eigenvalue corresponding to this eigenvector (1) (ii) hence show that p = 2 (2) (iii) determine the remaining eigenvalues and corresponding eigenvectors of A (7) (b) Write down a matrix P and a diagonal matrix D such that A = PDP −1 (1) (c) (i) Solve the differential equation u˙ = ku , where k is a constant. (2) With respect to a fixed origin O, the velocity of a particle moving through space is modelled by    x y z         = A x y z         By considering u v w         = P −1 x y z         so that    u v w         = P −1    x y z         (ii) determine a general solution for the displacement of the particle. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________