8.4. BASIC PROPERTIES OF THE DETERMINANT 177

= ∑(r1··· ,rn)

∑(k1,··· ,kn)

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

= ∑(r1··· ,rn)

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

Note that this shows that if two matrices are similar, then they have the same determi-nant and also the same characteristic polynomial, det(λ I−A).

8.4.1 Binet Cauchy FormulaThe Binet Cauchy formula is a generalization of the theorem which says the determinantof a product is the product of the determinants. The situation is illustrated in the followingpicture where A,B are matrices.

B A

Theorem 8.4.5 Let A be an n×m matrix with n≥ m and let B be a m×n matrix. Also letAi, i = 1, · · · ,C (n,m) be the m×m submatrices of A which are obtained by deleting n−mrows and let Bi be the m×m submatrices of B which are obtained by deleting correspondingn−m columns. Then

det(BA) =C(n,m)

∑k=1

det(Bk)det(Ak)

Proof: This follows from a computation. By Corollary 8.3.1 on Page 175, det(BA) =

1m! ∑

(i1···im)∑

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

1m! ∑

(i1···im)∑

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

n

∑r1=1

Bi1r1Ar1 j1

n

∑r2=1

Bi2r2 Ar2 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

=C(n,m)

∑k=1

∑{r1,··· ,rm}=Ik

1m! ∑

(i1···im)sgn(i1 · · · im)Bi1r1 Bi2r2 · · ·Bimrm ·

∑( j1··· jm)

sgn( j1 · · · jm)Ar1 j1Ar2 j2 · · ·Arm jm