Definition A.9.10A vector, w, is a linear combination of the vectors
{v ,⋅⋅⋅,v }
1 r
if there exist scalars c1,
⋅⋅⋅
crsuch thatw =∑k=1rckvk. This is the sameas saying w ∈span
(v ,⋅⋅⋅,v )
1 r
.
The following corollary is also of great use.
Corollary A.9.11Suppose A is an n×n matrix and some column (row) is a linearcombination of r other columns (rows). Then det
(A)
= 0.
Proof:Let A =
(a1 ⋅⋅⋅ an )
be the columns of A and suppose the condition that
one column is a linear combination of r of the others is satisfied. Say ai = ∑j≠icjaj. Then by
Corollary A.9.9, det(A) =
( ∑ ) ∑ ( )
det a1 ⋅⋅⋅ j⁄=icjaj ⋅⋅⋅ an = cjdet a1 ⋅⋅⋅ aj ⋅⋅⋅ an = 0
j⁄=i
because each of these determinants in the sum has two equal rows. ■
Recall the following definition of matrix multiplication.
Definition A.9.12If A and B are n × n matrices, A =
(aij)
and B =
(bij)
,AB =
(cij)
where cij≡∑k=1naikbkj.
One of the most important rules about determinants is that the determinant of a product
equals the product of the determinants.
Theorem A.9.13Let A and B be n × n matrices.Then
det(AB ) = det(A )det(B ).
Proof:Let cij be the ijth entry of AB. Then by Proposition A.9.6,
det(AB ) = ∑ sgn(k ,⋅⋅⋅,k )c ⋅⋅⋅c
(k,⋅⋅⋅,k ) 1 n 1k1 nkn
1 n ( ) ( )
∑ ∑ ∑
= sgn(k1,⋅⋅⋅,kn) r a1r1br1k1 ⋅⋅⋅ r anrnbrnkn
(k1,⋅⋅⋅,kn) 1 n
The Binet Cauchy formula is a generalization of the theorem which says the determinant
of a product is the product of the determinants. The situation is illustrated in the following
picture where A,B are matrices.
PICT
Theorem A.9.14Let A be an n × m matrix with n ≥ m and let B be a m × nmatrix. 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 Bibe them × m submatrices of B which are obtained by deleting corresponding n − m columns.Then
C(∑n,m )
det(BA ) = det(Bk )det(Ak )
k=1
Proof: This follows from a computation. By Corollary A.9.8 on Page 2161,
det