1.9. THE MATHEMATICAL THEORY OF DETERMINANTS 45

Theorem 1.9.13 Let A and B be n×n matrices. Then

det(AB) = det(A)det(B) .

Proof: Let ci j be the i jth entry of AB. Then by Proposition 1.9.6,

det(AB) = ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)c1k1 · · ·cnkn

= ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)

(∑r1

a1r1br1k1

)· · ·

(∑rn

anrnbrnkn

)

= ∑(r1··· ,rn)

∑(k1,··· ,kn)

sgn(k1, · · · ,kn)br1k1 · · ·brnkn (a1r1 · · ·anrn)

= ∑(r1··· ,rn)

sgn(r1 · · ·rn)a1r1 · · ·anrn det(B) = det(A)det(B) .■

The Binet Cauchy formula is a generalization of the theorem which says the deter-minant of a product is the product of the determinants. The situation is illustrated in thefollowing picture where A,B are matrices.

B A

Theorem 1.9.14 Let A be an n×m matrix with n≥ m and let B be a m×n matrix.Also let Ai

i = 1, · · · ,C (n,m)

be the m×m submatrices of A which are obtained by deleting n−m rows and let Bi bethe m×m submatrices of B which are obtained by deleting corresponding n−m columns.Then

det(BA) =C(n,m)

∑k=1

det(Bk)det(Ak)

Proof: This follows from a computation. By Corollary 1.9.8 on Page 43, det(BA) =

1m! ∑

(i1···im)∑

( j1··· jm)sgn(i1 · · · im)sgn( j1 · · · jm)(BA)i1 j1 (BA)i2 j2 · · ·(BA)im jm

=1

m! ∑(i1···im)

∑( j1··· jm)

sgn(i1 · · · im)sgn( j1 · · · jm) ·

n

∑r1=1

Bi1r1Ar1 j1

n

∑r2=1

Bi2r2Ar2 j2 · · ·n

∑rm=1

BimrmArm jm

Now denote by Ik one of the subsets of {1, · · · ,n} which has m elements. Thus there areC (n,m) of these.

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m! ∑

(i1···im)∑

( j1··· jm)sgn(i1 · · · im)sgn( j1 · · · jm) ·

Bi1r1 Ar1 j1Bi2r2Ar2 j2 · · ·BimrmArm jm