Your Practice Set – Applications and Interpretation for IBDP Mathematics
3 . A district is served by two supermarkets , A and B . Customers living in the district can choose buying in either A or B . Each week 82 % of the customers choosing A and 87 % of the customers choosing B will not change their choices in the next week . Assume that there is no net gain or net loss of the total number of customers .
( a ) Find T , the transition matrix representing the changes in the number of customers choosing supermarkets in a particular week .
( 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 .
[ 2 ]
[ 3 ]
( c ) Write down v
1
and v
2
.
It is given that matrix . n n �1
T � PD P , where P is a 2� 2 matrix and D is a 2� 2 diagonal
[ 2 ]
( d ) Write down
( i ) P ;
( ii ) n
D .
( e ) Hence , express n
T in terms of n .
The initial number of customers choosing each supermarket are both 310 .
( f ) Write down
� 310�
T n � �.
�310�
( g ) Hence , write down , after a long term , the numbers of customers choosing
[ 3 ]
[ 3 ]
[ 2 ]
( i ) A ;
( ii ) B . [ 2 ]
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