Page 34 - Matrices theory and applications

2.1. Determinants and Minors
called a minor of order p. Once the choice of the row indices i 1 < ··· <i p
and column indices j 1 < ··· <j p has been made, one denotes by

i 2
i 1
M
···
j 1
j p
the corresponding minor. A principal minor is a minor with equal row and
column indices, that is, of the form j 2 ··· i p 17

i 1 i 2 ··· i p
M .
i 1 i 2 ··· i p
In particular, the leading principal minor of order p is

12 ··· p
M .
12 ··· p
ˆ
Given a matrix M ∈ M n(A), one associates the matrix M of cofactors,
defined as follows: its (i, j)-th entry ˆm ij is the minor of order n−1 obtained
by removing the ith row and the jth column multiplied by (−1) i+j .Itis
also the factor of m ij in the formula for the determinant of M. Finally, we
define the adjoint matrix adj M by
ˆ T
adj M := M .
Proposition 2.1.1 If M ∈ M n (A), one has
M(adj M)= (adj M)M =det M · I n . (2.1)
Proof
The identity is clear as far as diagonal terms are concerned; it amounts to
the definition of the determinant (see also below). The off-diagonal terms
m ij of M(adj M) are sums involving on the one hand an index, and on
the other hand a permutation σ ∈ S n . One groups the terms pairwise,
corresponding to permutations σ and στ,where τ is the tranposition (i, j).
The sum of two such terms is zero, so that m =0.

ij
Proposition 2.1.1 contains the well-known and important expansion for-
mula for the determinant with respect to either a row or a column. The
expansion with respect to the ith row is written
det M =(−1) i+1 m i1 ˆm i1 + ··· +(−1) i+n m in ˆm in ,

while the expansion with respect to the ith column is
det M =(−1) i+1 m 1i ˆm 1i + ··· +(−1) i+n m ni ˆm ni .

2.1.1 Irreducibility of the Determinant
By definition, the determinant is a polynomial function, in the sense that
det M is the value taken by a polynomial Det A ∈ A[x 11 ,... ,x nn ]when the