4 . The proportion of a group of people supporting the political parties , A and B , are studied . Each year 10 % of the people supporting A and 7 % of the people supporting B will change their preferences in the next year . It is assumed that people can either choose supporting A or B , and there is no change in the number of people in the group .
( a ) Find T , the transition matrix representing the changes in the proportion of political party supporters in a particular year .
( b ) Find the values of �
1
and �
2
, the eigenvalues of T , where �1 � �2.
Let v
1
and v
2 be the eigenvectors of T corresponding to �
1
and �
2 respectively .
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( c ) Write down v
1
and v
2
.
It is given that matrix .
( d ) Write down
n n �1
T � PD P , where P is a 2� 2 matrix and D is a 2� 2 diagonal
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( i ) P ;
( ii ) n
D .
( e ) Hence , express n
T in terms of n .
It is given that the ratio of the number of supporters of A to that of B is 3:2 in 2019 .
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( f ) |
0.6 Write down
� T n �
�.
�0.4�
|
( g ) |
Hence , find the proportion of the supporters of A in 2013 . |
( h ) |
Using ( f ) to write down , after a long term , the proportions of supporters of |
( i ) |
A ; |
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( ii ) B . [ 2 ] www . seprodstore . com
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