ABSTRACT
Definition. The square array (matrix) A, with n rows and n columns, has associated with it the determinant
detA
a a a a a a
a a a
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
,
a number equal to
∑ ±( )a a a ai j k nl1 2 3 where i, j, k, … ,l is a permutation of the n integers 1, 2, 3, … , n in some order. The sign is plus
if the permutation is even and is minus if the permutation is odd (see Section 1.12). The 2 × 2 determinant
a a a a 11 12
has the value a11a22−a12a21 since the permutation (1, 2) is even and (2, 1) is odd. For 3 × 3 determinants, permutations are as follows:
1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1
, , , , , , , , , , , ,
even odd odd even even odd
Thus,
a a a a a a a a a
a a a a a a a a11 12 13
12 =
+ ⋅ ⋅
− ⋅ ⋅
− ⋅ 21 33
⋅
+ ⋅ ⋅
+ ⋅ ⋅
− ⋅ ⋅
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
a a a a a a a a a a
⎪
⎪
⎭
⎪
⎪
⎪
⎪
A determinant of order n is seen to be the sum of n! signed products.
