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APPENDIX 2 DETERMINANTS
where
D 1 ˆ b1 a 12
b 2 a 22 ; D 2 ˆ a11 b 1
a 21 b 2 ; D ˆ a11 a 12
a 21 a 22
…A2:4†
are called determinants of second order or order 2. The numbers enclosed between
vertical bars are called the elements of the determinant. The elements in a
horizontal line form a row and the elements in a vertical line form a column
of the determinant. It is obvious that in Eq. (A2.3) D 6ˆ 0.
Note that the elements of determinant D are arranged in the same order as they
occur as coecients in Eqs. (A1.1). The numerator D 1 for x 1 is constructed from
D by replacing its ®rst column with the coecients b 1 and b 2 on the right-hand
side of (A2.1). Similarly, the numerator for x 2 is formed by replacing the second
column of D by b 1 ; b 2 . This procedure is often called Cramer's rule.
Comparing Eqs. (A2.3) and (A2.4) with Eq. (A2.2), we see that the determinant
is computed by summing the products on the rightward arrows and subtracting
the products on the leftward arrows:
a 11 a 12
a 21 a 22
ˆ a 11a 22 a 12 a 21 ; etc:
…†
…‡†
This idea is easily extended. For example, consider the system of three linear
equations
9
a 11 x 1 ‡ a 12 x 2 ‡ a 13 x 3 ˆ b 1 ; >=
a 21 x 1 ‡ a 22 x 2 ‡ a 23 x 3 ˆ b 2 ;
>;
a 31 x 1 ‡ a 32 x 2 ‡ a 33 x 3 ˆ b 3 ;
in three unknowns x 1 ; x 2 ; x 3 . To solve for x 1 , we multiply the equations by
a 22 a 33 a 32 a 23 ; …a 12 a 33 a 32 a 13 †; a 12 a 23 a 22 a 13 ;
respectively, and then add, ®nding
…A2:5†
x 1 ˆ b1a 22 a 33 b 1 a 23 a 32 ‡ b 2 a 13 a 32 b 2 a 12 a 33 ‡ b 3 a 12 a 23 b 3 a 13 a 22
a 11 a 22 a 33 a 11 a 32 a 23 ‡ a 21 a 32 a 13 a 21 a 12 a 33 ‡ a 31 a 12 a 23 a 31 a 22 a 13
;
which can be written in determinant form
x 1 ˆ D 1 =D;
…A2:6†
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