Question 3:
  a) 2 marks  for eigenvaluess  1, -1, -3
(quite a few got this wrong - they got 1, 1, -3)

  b) 1 and 1 marks: for  det=3 and trace=-3
(quite a few missed this - most did it the long way rather than use eigs.)

  c) 4 marks for basis of eigenvectrs:
      e.g. (1 0 1)   (-1 1 -1)   (0 1 1)
(Surprisingly many missed the simple GE to find these 3 eigenvectors correctly.
I did not see anyone 'check' their arithmetic to ensure they had
eigenvectors. Quite a few used a 0 eigenvector.)

      3/4 if applied Gram-Schmidt unnecessarily
      2/4 if arith mistake and one vector is wrong
      1/4 if two vectors wrong or got a zero vector

  d) 2 marks for general solution of ODE statement
(Almost everyone did this well.)

  e) 2 marks to solve system and get IVP e.g. (1 -1 1)
       (I did not expect them to write down actual general solution just
         need to find constants. Most people did not write down the
IVP solution and many wrote the wrong statement.)


Question 4:

  a)   3 and 3 and 3 for eigs, alg mult, geom mult
      eigs:2,1,3 alg. mult: 3,2,3  geom. mult: 1,2,1
       7/9 if all right except g_1=1 rather than g_1=2
  
  b)   3 marks for NOT DZ with explanation! (0 without explanation)
      (not DZ since geom and alg mult not the same)
       2/3 marks if explanation but did not understand geom mult in a)
   (I had to make a judgement whether student understood how
to find the geometric mult.)
       0/3 if wrong or no answer for DZ, i.e. just explanation does not count