ABSTRACT
Denition. The square array (matrix) A, with n rows and n columns, has associated with it the determinant
det A
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 1.12). The 2 × 2 determinant
a a
a a
has the value a a a a11 22 12 21− since the permutation (1, 2) is even and (2, 1) is odd. For 3 × 3 determinants, permutations are as follows:
1, 2, 3 even
1, 3, 2 odd
2, 1, 3 odd
2, 3, 1 even
3, 1, 2 even
3, 2, 1 odd
Thus,
a a a
a a a
a a a
a a a
=
+
−
. .
