Let �
1
and �
2 be the eigenvalues of M , where �1 � �2.
( a ) Find det ( M��I ) , giving the answer in terms of � .
( 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 .
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( c ) Write down v
1
and v
2
.
( d ) Hence , write down the general solution for
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( i ) x ;
( ii ) y . [ 2 ]
3 . Consider the coupled differential equations
matrix equation
two 2� 1 matrices .
X � MX , where M is a 2� 2 matrix , and
�dx
�4x�2y
� dt �� . It can be expressed by a dy
� �9x� 2y
�� dt �dx �
� dt �
X � � � and dy � � � dt �
�x �
X � � � are �y �
( a ) Find det ( M��I ) , giving the answer in terms of � .
( b ) Hence , write down the values of the eigenvalues of M .
Let v
1
and v
2 be the eigenvectors of M .
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( c ) Write down the general solution for X .
The phase portrait for the coupled differential equation is in the shape of spiral .
( d ) Describe the stability of the phase portrait .
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