Page 16 - Determinants and Their Applications in Mathematical Physics

1

Determinants, First Minors, and
Cofactors

1.1 Grassmann Exterior Algebra

Let V be a finite-dimensional vector space over a field F. Then, it is known
that for each non-negative integer m, it is possible to construct a vector
0
space Λ V . In particular, Λ V = F,ΛV = V , and for m ≥ 2, each vector
m
in Λ V is a linear combination, with coefficients in F, of the products of
m
m vectors from V .
If x i ∈ V ,1 ≤ i ≤ m, we shall denote their vector product by x 1 x 2 ··· x m .
Each such vector product satisfies the following identities:

i. x 1 x 2 ··· x r−1 (ax + by)x r+1 ··· x n = ax 1 x 2 ··· x r−1 xx r+1 ··· x n
+bx 1 x 2 ··· x r−1 y ··· x r+1 ··· x n , where a, b ∈ F and x, y ∈ V .
ii. If any two of the x’s in the product x 1 x 2 ··· x n are interchanged, then
the product changes sign, which implies that the product is zero if two
or more of the x’s are equal.

1.2 Determinants

Let dim V = n and let e 1 , e 2 ,..., e n be a set of base vectors for V . Then,
if x i ∈ V ,1 ≤ i ≤ n, we can write

n

x i = a ik e k , a ik ∈ F. (1.2.1)
k=1