Determinants of order 2 and 3 were defined in Chapter 2 by the formulae
The similarity in form between determinants
and matrices suggests a close relationship between them.
where
There are many different, but equivalent, ways to carry out the generalization. One way that emphasizes the anti-symmetry of the determinant function is to define it as
where P
is a permutation of n integers, i.e.,
In general, there are n! terms in the sum. For
example, the case
P_{1} |
P_{2} |
P_{3} |
(-)^{P} |
1 |
2 |
3 |
+1 |
1 |
3 |
2 |
-1 |
2 |
1 |
3 |
-1 |
2 |
3 |
1 |
+1 |
3 |
1 |
2 |
+1 |
3 |
2 |
1 |
-1 |
so that
as can be checked against (5.2) with the help of (5.1).
On the other hand, there is also a
geometrical interpretation of a determinant as a volume in