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Col. Col.
1 2
–––––––
�a1 b1� Determinant � D �
�a
� a1b2 � a2b1 2 b2� (C.3)
The expanded value is obtained by first multiplying the top left element
by the bottom right and then subtracting the product of the lower
left and upper right elements. This particular determinant is referred to
as a second-order determinant, since it contains two rows and two
columns.
It is important to remember when using determinants that the
columns of the equations, as indicated in Eqs. (C.1a) and (C.1b), must
be placed in the same order within the determinant configuration. That
is, since a 1 and a 2 are in column 1 of Eqs. (C.1a) and (C.1b), they must
be in column 1 of the determinant. (The same is true for b 1 and b 2.)
Expanding the entire expression for x and y, we have the following:
EXAMPLE C.1 Evaluate the following determinants:
�2 2�
a. � (2)(4) � (3)(2) � 8 � 6 � 2
�3 4�
�4 �1�
b. � (4)(2) � (6)(�1) � 8 � 6 � 14
�6 �2�
� 0 �2�
c. � (0)(4) � (�2)(�2) � 0 � 4 � �4
��2 �4�
�0 10�
d. � (0)(10) � (3)(0) � 0
�3 10�
�c 1 b 1�
�c 2 b 2 � c 1b 2 � c 2b 1
x � ––––––– � ––––––––––
�a 1 b 1� a 1b 2 � a 2b 1
�a 2 b 2 �
�a 1 c 1�
�a 2 c 2 � a 1c 2 � a 2c 1
y � ––––––– � ––––––––––
�a 1 b 1� a 1b 2 � a 2b 1
�a 2 b 2 �
EXAMPLE C.2 Solve for x and y:
2x � y � 3
3x � 4y � 2
Solution:
�3 1�
�2 4� (3)(4) � (2)(1) 12 � 2 10
x ����� ��� � � 2
�2 1� (2)(4) � (3)(1) 8 �3 5
�3 4�
(C.4a)
(C.4b)
APPENDIX C ⏐⏐⏐ 1199