The characteristic equation is det a i 9 0 4 0 det λ

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The characteristic equation is:det(AI)=9.04.0det=λ20.36=(λ –0.6) (λ + 0.6) = 0.The eigenvalues are= 0.6 and=0.6 .The corresponding eigenvectors are the solutions of(AI)x=0.= 0.66.09.04.06.06.0IA32x=0.66.09.04.06.06.0IA32xWrite the starting vector as a linear combination of the eigenvectors and applyAntimes:323201baThis solves to givea= 1/4andb=1/4:3241324101Now applyAn:32)6.0(4132)6.0(4101nnnnnAyxWrite as two separate equations:nnnx6.0216.021nnny6.0436.043.page67
MATH 111 April Exam Solutions 2016page 9 of 11 pagesXYE0.90.10.60.4(d) [3 marks] Using the node diagram and elementary reasoning (as ifyou were in high school but understood the diagram), work outdirectly (without any linear algebra) the probabilitiesxnandynthat acounter starting on node X will be on nodes X and Y after exactlynsteps.[For example, you should be able to see exactly what has tohappen for a counter on X to find itself on Y after exactly 5 moves.]Starting on X, a counter which does not get to E will simply cyclebetween X and Y.It will be on Y after all odd numbers of moves andon X after all even numbers of moves.The probability of a cycle is(0.9) (0.4) = 0.36.Thusodd0even)36.0(2/nnxnneven0odd)36.0)(9.0(2/)1(nnynn(e) [3 marks] Now simplify your answers to parts (c) and check that they agree with your answerto (d).nnnx6.0216.021neven:2/2/236.06.06.06.0216.0216.0)1(216.021nnnnnnnnnxnodd:06.0216.0216.0)1(216.021nnnnnnxnnny6.0436.043.nodd:16.0)6.0(236.0236.0436.0436.0)1(436.043nnnnnnnny2/)1(2/)1(236.0)9.0(6.0)9.0(nnneven:06.0436.0436.0)1(436.043nnnnnny.page68
MATH 111 April Exam Solutions 2016page 10 of 11 pagesWG1/41/43/41/41/41/45. [16 marks] At the right is a section of an infinite grid––enough has been diagrammed to make the pattern clear.Some nodes are white (W) and some are grey (G).Twonodes are “neighbours” if they are connected by an edge.Suppose that the nodes are inhabited by apopulation of counters.Some nodes might bevacant, others will be occupied by one or morecounters.Read the following rules carefully as theanswers to all the questions below will depend onthem.In each time unit, the counters behave asfollows.A counter on a W-node.moves to one of its W-neighbours with probability 3/4moves to its G-neighbour with probability 1/4A counter on a G-node.

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The Land, Probability theory, Eigenvalue eigenvector and eigenspace, Det

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