Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
� Operations of matrices :
1 .
2 .
� a11 a1 n � � b11 b1 n � � a11 �b11 a1 n �b1 n � � � � � � �
�
� � a a � �b b � � a �b a �b
� � m1 mn � � m1 mn � � m1 m1 mn mn �
� a11 a1 n � � ka11 ka1 n � � � � � k
� � a a � � ka ka � � m1 mn � � m1 mn �
3 . c � a
1b1 � a
2b2
� � a b : The element on the i th row and the j th ij i j i j in nj
column of
� A 2� 2 system
solved by
� A 3� 3 system
be solved by
� a11 a1 n ��b11 b1 k � � �� �
C �AB � �� , where A , B and C � a a ��b b � � m1 mn �� n1 nk � are m� n, n� k and m� k matrices respectively
�ax �by �c � �dx �ey � f can be expressed as
�1
�1 � a b� � c �
X �A B � � � � � � d e� � f �
�ax � by � cz � d � �ex � fy � gz � h can be expressed as � �ix � jy � kz � l
X A B
�1
� a b c � � d � � � � �
�1 � � � e f g � � h � � i j k � � l � � � � �
� Eigenvalues and eigenvectors of A : 1 . det ( A��I ) : Characteristic polynomial of A 2 . Solution ( s ) of det ( A��I ) �0
: Eigenvalue ( s ) of A
AX � B , where
AX � B , where
�x
�
X � � � can be �y
�
�x
� � �
X � y can �z
� � �
3 . v : Eigenvector of A corresponding to the eigenvalue � , which satisfies Av � �v
� Diagonalization of A :
1 .
|
��1
�
0
D
�
� �
|
0
�2
|
0
�
�
0
�: Diagonal matrix of the eigenvalues of A �
|
|
�
�
0
|
0 |
�
�n
�
|
2 . � � � 3 .
V v v v : A matrix of the eigenvectors of A
1 2 n
A � VDV � A � VD V
�1 n n �1
16
SE Production Limited