Page 32 - Matrices theory and applications

2
Square Matrices

The essential ingredient for the study of square matrices is the determinant.
For reasons that will be given in Section 2.5, as well as in Chapter 6, it
is useful to consider matrices with entries in a ring. This allows us to
consider matrices with entries in ZZ (rational integers) as well as in K[X]
(polynomials with coefficients in K). We shall assume that the ring A of
scalars is a commutative (meaning that the multiplication is commutative)
integral domain (meaning that it does not have zero divisors: ab = 0 implies
either a =0 or b = 0), with a unit denoted by 1, that is, an element
satisfying 1x = x1= x for every x ∈ A. Observe that the ring M n(A)is
not commutative if n ≥ 2. For instance,

0 1 0 0 1 0 0 0 0 0 0 1
= = = .
0 0 1 0 0 0 0 1 1 0 0 0
An element a of A is invertible if there exists b ∈ A such that ab =1.
The element b is unique (because A is an integral domain), and one calls it
the inverse of a, with the notation b = a −1 . The set of invertible elements
of A is a multiplicative group, denoted by A . One has
∗
(ab) −1 = b −1 −1 = a −1 −1 .
a
b

2.1 Determinants and Minors

We recall that S n ,the symmetric group, denotes the group of permutations
over the set {1,... ,n}.