Page 35 - Matrices theory and applications

2. Square Matrices
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x ij ’s are replaced by the scalars m ij .We observe that Det A does not really
depend on the ring A, in the sense that it is the image of Det ZZ through
the canonical ring homomorphism ZZ → A. For this reason, we shall simply
write Det. The polynomial Det may be viewed as the determinant of the
matrix X =(x ij ) 1≤i,j≤n ∈ M n (A[x 11 ,... ,x nn ]).
Theorem 2.1.1 The polynomial Det is irreducible in A[x 11 ,... ,x nn ].
Proof
We shall proceed by induction on the size n.If n = 1, there is nothing
to prove. Thus let us assume that n ≥ 2. We denote by D the ring of
polynomials in the x ij with (i, j) =(1, 1), so that A[x 11 ,... ,x nn ]= D[x 11 ].
From the expansion with respect to the first row, we see that Det = x 11 P +
Q,with P, Q ∈ D. Since Det is of degree one as a polynomial in x 11 ,
any factorization must be of the form (x 11 R + S)T ,with R, S, T ∈ D.In
particular, RT = P.
By induction, and since P is the polynomial Det of (n − 1) × (n − 1)
matrices, it is irreducible in E, the ring of polynomials in the x ij ’s with
i, j > 1. Therefore, it is also irreducible in D,since D is the polynomial
ring E[x 12 ,... ,x 1n ,x 21 ,... ,x n1 ]. Therefore, we may assume that either R
or T equals 1.
If the factorization is nontrivial, then R =1 and T = P. It follows that
P divides Det. An expansion with respect to variousrowsshows similarly
that every minor of size n− 1, considered as an element of A[x 11 ,... ,x nn ],
divides Det. However, each such minor is irreducible, and they are pairwise
distinct, since they do not depend on the same set of x ij ’s. We conclude
that the product of all minors of size n − 1 divides Det. In particular, the
2
degree n of Det is greater than or equal to the degree n (n − 1) of this
product, an obvious contradiction.

2.1.2 The Cauchy–Binet Formula

In the sequel, we shall use also the following result.
Proposition 2.1.2 Let B ∈ M n×m (A), C ∈ M m×l (A), and an integer
p ≤ n, l be given. Let 1 ≤ i 1 < ··· <i p ≤ n and 1 ≤ k 1 < ··· <k p ≤ l be
indices. Then the minor

i 1 i 2 ··· i p
(BC)
k 1 k 2 ··· k p
is given by the formula

i 1 i 2 ··· i p j 1 j 2 ··· j p
B · C .
j 1 j 2 ··· j p k 1 k 2 ··· k p
1≤j 1 <j 2 <···<j p ≤m