Linear Algebra, Theory And Applications, 2012a

3.3. THE MATHEMATICAL THEORY OF DETERMINANTS 89
The following corollary is also of great use.
Corollary 3.3.11 Suppose A is an n × n matrix and some column (row) is a linear combination
of r other columns (rows). Then det (A) =0.
Proof: Let A = ( a 1 ··· a n
)
be the columns of A and suppose the condition that
one column is a linear combination of r of the others is satisfied. Then by using Corollary
3.3.9 you may rearrange the columns to have the n th column a linear combination of the
first r columns. Thus a n = ∑ r
k=1 c ka k and so
By Corollary 3.3.9
det (A) =det ( a 1 ··· a r ··· a n−1
∑ r
k=1 c ka k
)
.
det (A) =
r∑
c k det ( )
a 1 ··· a r ··· a n−1 a k =0.
k=1
The case for rows follows from the fact that det (A) =det ( A T ) .
Recall the following definition of matrix multiplication.
Definition 3.3.12 If A and B are n × n matrices, A =(a ij ) and B =(b ij ), AB =(c ij )
where c ij ≡ ∑ n
k=1 a ikb kj .
One of the most important rules about determinants is that the determinant of a product
equals the product of the determinants.
Theorem 3.3.13 Let A and B be n × n matrices. Then
det (AB) =det(A)det(B) .
Proof: Let c ij be the ij th entry of AB. Then by Proposition 3.3.6,
det (AB) =
∑
sgn (k 1 , ··· ,k n ) c 1k1 ···c nkn
=
=
=
∑
(k 1 ,··· ,k n )
∑
(k 1 ,··· ,k n )
(r 1··· ,r n ) (k 1 ,··· ,k n )
∑
(r 1··· ,r n )
( ) ( )
∑ ∑
sgn (k 1 , ··· ,k n ) a 1r1 b r1k 1
··· a nrn b rnk n
r 1 r n
∑
sgn (k 1 , ··· ,k n ) b r1 k 1
···b rn k n
(a 1r1 ···a nrn )
sgn (r 1 ···r n ) a 1r1 ···a nrn det (B) =det(A)det(B) .
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.
B
A
Theorem 3.3.14 Let A be an n × m matrix with n ≥ m and let B be a m × n matrix. Also
let A i
i =1, ··· ,C(n, m)