Page 33 - Matrices theory and applications

2. Square Matrices
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Let M ∈ M n(A) be a square matrix. Its determinant is defined by

(σ)m 1σ(1) ··· m nσ(n) ,
det M :=
σ∈S n
where the sum ranges over all the permutations of the integers 1,... ,n.
We denote by (σ)= ±1 the signature of σ,equal to +1 if σ is the product

an even number of transpositions, and −1 otherwise. Recall that (σσ )=

(σ) (σ ).
If M is triangular, then all the products vanish other than the one
associated with the identity (that is, σ(j)= j). The determinant of a
triangular M is thus equal to the product of diagonal entries m ii .In par-
ticular, det I n =1and det 0 n = 0. An analogous calculation shows that
the determinant of a block triangular matrix is equal to the product of the
determinants of the diagonal blocks M jj .
Since (σ −1 )= (σ), one has
det M T =det M.
n
Looking at M as a row matrix with entries in A , one may view the
determinant as a multilinear form of the n columns of M:

det M =det M (1) ,... ,M (n) .
This form is alternate: If two columns are equal, the determinant vanishes.
As a matter of fact, if the ith and the jth columns are equal, one groups the
permutations pairwise (σ, τσ), where τ is the transposition (i, j). For each
pair, both products are equal, up to the signatures, which are opposite;
their sum is thus zero. Likewise, if two rows are equal, the determinant is
zero.
More generally, if the columns of M satisfy a non trivial linear relation
(a 1 ,... ,a n not all zero) of linear dependence
a 1 M 1 + ··· + a n M n =0
(that is, if rk M< n), then det M is zero. Let us assume, for instance, that
a 1 is nonzero. For j ≥ 2, one has

det M (j) ,M (2) ,... ,M (n) =0.
Using the multilinearity, one has thus

a 1 det M = det a 1 M (1) + ··· + a n M (n) ,M (2) ,... ,M (n)

= det 0,M (2) ,... =0.
Since A is an integral domain, we conclude that det M =0.
For a matrix M ∈ M n×m(A), not necessarily square, and p ≥ 1 an integer

with p ≤ m, n,one may extract a p × p matrix M ∈ M p (A) by retaining

only p rows and p columns of M. The determinant of such a matrix M is