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APPENDIX 2 DETERMINANTS
where
a 11 a 12 a 13
b 1 a 12 a 13
D ˆ
a 21 a 22 a 23
; D 1 ˆ
b 2 a 22 a 23
: …A2:7†
a 31 a 32 a 33
b 3 a 32 a 33
Again, the elements of D are arranged in the same order as they appear as
coecients in Eqs. (A2.5), and D 1 is obtained by Cramer's rule. In the same
manner we can ®nd solutions for x 2 ; x 3 . Moreover, the expansion of a determinant
of third order can be obtained by diagonal multiplication by repeating on the
right the ®rst two columns of the determinant and adding the signed products of
the elements on the various diagonals in the resulting array:
a 11 a 12 a 13
a 11 a 12
a 21 a 22 a 23
a 21 a 22
a 31 a 32 a 33
a 31 a 32
…† …† …†
…‡† …‡† …‡†
This method of writing out determinants is correct only for second- and thirdorder
determinants.
Problem A2.1
Solve the following system of three linear equations using Cramer's rule:
2x 1 x 2 ‡ 2x 3 ˆ 2;
x 1 ‡ 10x 2 3x 3 ˆ 5;
x 1 ‡
x 2 ‡ x 3 ˆ3:
Problem A2.2
Evaluate the following determinants
5 1 8
1 2
…a†
4 3 ; …b† 15 3 6
; …c†
10 4 2
cos
sin
sin
cos :
Determinants, minors, and cofactors
We are now in a position to de®ne an nth-order determinant. A determinant of
order n is a square array of n 2 quantities enclosed between vertical bars,
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