Your Practice Set – Applications and Interpretation for IBDP Mathematics
Exercise 29
1 . The matrix A is defined by
where �1 � �2.
��2 �3�
A � � � . Let �
1
and �
2 be the eigenvalues of A ,
� 1 2 �
( a ) Find the characteristic polynomial of A .
( b ) Hence , write down the values of �
1
and �
2
.
Let v
1
and v
2 be the eigenvectors of A corresponding to �
1
and �
2 respectively .
[ 2 ]
[ 2 ]
( c ) Write down v
1
and v
2
.
It is given that
( d ) Find � .
It is given that matrix .
� det ( A ) � , where � � . ��
1 2
n n �1
A � PD P , where P is a 2� 2 matrix and D is a 2� 2 diagonal
[ 2 ]
[ 2 ]
( e ) Write down
( i ) P ;
( ii ) n
D .
( f ) Hence , express n
A in terms of n .
[ 3 ]
[ 3 ]
2 . The matrix A is defined by
where �1 � �2.
�9 �4�
A � � � . Let �
1
and �
2 be the eigenvalues of A ,
�2 3 �
( a ) Find det ( A��I ) , giving the answer in terms of � .
( b ) Hence , write down the values of �
1
and �
2
.
[ 2 ]
[ 2 ]
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