Your Practice Set – Applications and Interpretation for IBDP Mathematics
( g ) n n
� 2 � 1 � 3 3 � 1 � 3 � � � � � � � � � � n
5 � 4 � 5 5 � 4 � 5 T �
� �
A1 � n
n � 2 � 1 � 2 3 � 1 � 2
�� � � � � � � 5 4 5 5 4 5
� � � � � � �
12 12
� 2 � 1 � 3 3 � 1 � 3 � �
550
� � � � � � � � � � � 5 � 4 � 5 5 � 4 � 5 ��550�
� 450� �
� 2 1 2 3 1 2 450 � � � � � �
�� � � � � � � 5 4 5 5 4 5
� � � � � � �
12 �550� �599.999997
�
T � � � � � � 450� � 400.000003 �
Thus , the population of X and Y after a year are 600 and 400 respectively . A2
12
T � � �
12 12 � � M1A1
[ 3 ]
[ 4 ]
Exercise 61
1 . In a town , citizens purchase computers from either Japanese manufacturers or American manufacturers . Each year 25 % of the citizens purchasing computers from American manufacturers change their choices to Japanese manufacturers , and 70 % of the citizens purchasing computers from Japanese manufacturers do not change their choices . Assume that there is no net gain or net loss of the entire population in the town .
( a ) Find T , the transition matrix representing the changes in the number of citizens choosing manufacturers in a particular year .
[ 2 ]
( b ) Find the characteristic polynomial of T . [ 2 ]
( c ) Hence , write down the values of �
1
and �
2
, the eigenvalues of T , where �1 � �2. [ 2 ]
Let v
1
and v
2 be the eigenvectors of T corresponding to �
1
and �
2 respectively .
( d ) 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 ]
( e ) Write down
( i ) P ;
( ii ) n
D . [ 3 ]
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