Page 30 - Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 15

2.3.6 The Cofactors of a Zero Determinant
If A = 0, then
, (2.3.16)
A p 1 q 1 A p 2 q 2 = A p 2 q 1 A p 1 q 2
that is,

A p 1 q 1

A p 1 q 2 =0, 1 ≤ p 1 ,p 2 ,q 1 ,q 2 ≤ n.
A p 2 q 1 A p 2 q 2

It follows that

A p 1 q 1 A p 1 q 2
A p 1 q 3
=0
A p 2 q 1 A p 2 q 2

A p 2 q 3

A p 3 q 1 A p 3 q 2 A p 3 q 2
since the second-order cofactors of the elements in the last (or any) row are
all zero. Continuing in this way,

A p 1 q 1 A p 1 q 2 ··· A p 1 q r

A p 2 q 1 A p 2 q 2 ··· A p 2 q r
=0, 2 ≤ r ≤ n. (2.3.17)

··· ··· ··· ···

A p r q 1 A p r q 2 ··· A p r q r r
This identity is applied in Section 3.6.1 on the Jacobi identity.
2.3.7 The Derivative of a Determinant
If the elements of A are functions of x, then the derivative of A with respect
to x is equal to the sum of the n determinants obtained by differentiating
the columns of A one at a time:
n

A =
C 1 C 2 ··· C ··· C n
j
j=1
n n

= a A ij . (2.3.18)

ij
i=1 j=1