1

Determinants, First Minors, and

Cofactors

1.1 Grassmann Exterior Algebra.................

LetVbe a finite-dimensional vector space over a fieldF. Then, it is known

that for each non-negative integerm, it is possible to construct a vector

space Λ

m

V. In particular, Λ

0

V=F,ΛV=V, and form≥2, each vector

in Λ

m

Vis a linear combination, with coefficients inF, of the products of

mvectors fromV.

Ifxi∈V,1≤i≤m, we shall denote their vector product byx 1 x 2 ···xm.

Each such vector product satisfies the following identities:

i. x 1 x 2 ···xr− 1 (ax+by)xr+1···xn=ax 1 x 2 ···xr− 1 xxr+1···xn

+bx 1 x 2 ···xr− 1 y···xr+1···xn, wherea, b∈Fandx,y∈V.

ii.If any two of thex’s in the productx 1 x 2 ···xnare interchanged, then

the product changes sign, which implies that the product is zero if two

or more of thex’s are equal.

1.2 Determinants..........................

Let dimV =nand lete 1 ,e 2 ,...,enbe a set of base vectors forV. Then,

ifxi∈V,1≤i≤n, we can write

xi=

n

∑

k=

aikek,aik∈F. (1.2.1)