7 . The matrix M is defined by
of M , where �1 � �2.
� 5 4 � �
3 3
� M � � �. Let �
1 and �
2 be the eigenvalues
2 1 � � � � � 3 3�
( a ) Find the characteristic polynomial of M .
( b ) Hence , write down the values of �
1 and �
2
.
Let v
1 and v
2 be the eigenvectors of M corresponding to �
1 and �
2 respectively .
( c ) Write down v
1 and v
2
. n n �1
It is given that M � PD P , where P is a 2� 2 matrix and D is a 2� 2 diagonal matrix .
[ 2 ]
[ 2 ]
[ 2 ]
( d ) Write down
( i ) P ;
( ii ) n
D .
( e ) Hence , express n
M in terms of n .
Let gn ( ) be the first diagonal entry of n
M .
[ 3 ]
[ 3 ]
( f ) Write down lim gn ( ) . n��
[ 1 ]
© SE Production Limited 26 All Rights Reserved 2021