Page 36 - Matrices theory and applications

2.2. Invertibility
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Corollary 2.1.1 Let b, c ∈ A.If b divides every minor of order p of B
and if c divides every minor of order p of C,then bc divides every minor
of order p of BC.
The particular case l = m = n is fundamental:
Theorem 2.1.2 If B, C ∈ M n (A),then det(BC)= det B · det C.
In other words, the determinant is a multiplicative homomorphism from
M n (A)to A.
Proof
The corollaries are trivial. We only prove the Cauchy–Binet formula.
Since the calculation of the ith row (respectively the jth column) of BC
involves only the ith row of B (respectively the jth column of C), one
may assume that p = n = l. The minor to be evaluated is then det BC.If
m<n, there is nothing to prove, since on the one hand the rank of BC
is less than or equal to m,thusdet BC is zero, and on the other hand the
left-hand side sum in the formula is empty.
There remains the case m ≥ n. Let us write the determinant of a ma-
trix P as that of its columns P j and let us use the multilinearity of the
determinant:
 
n

det BC = det  c j 1 1 B j 1 , (BC) 2 ,... , (BC) n 
j 1 =1
 
n n

= c j 1 1 det  B j 1 , c j 2 2 B j 2 , (BC) 3 ,... , (BC) n 
j 1 =1 j 2 =1

= ··· = c j 1 1 ··· c j n n det(B j 1 ,... ,B j n ).
1≤j 1 ,... ,j n ≤n
In the sum the determinant is zero as soon as f → j f is not injective,
since then there are two identical columns. If on the contrary j is injective,
this determinant is a minor of B, up to the sign. This sign is that of the
permutation that puts j 1 ,... ,j p in increasing order. Grouping in the sum
the terms corresponding to the same minor, we find that det BC equals

1 2 ··· n
(σ)c k 1 σ(1) ··· c k n σ(n) B ,
k 1 k 2 ··· k n
1≤k 1 <···<k n ≤m,
σ∈S n
which is the required formula.

2.2 Invertibility

Since M n (A) is not an integral domain, the notion of invertible elements
of M n (A) needs an auxiliary result, presented below.