Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 399

This result may be verified by selecting the values of mnp to be 123, 231, 312,
132, 321, or 213, then performing the summation over the three indices i , j , and k
over 1, 2, and 3 as indicated on the right-hand side of (A.107). In each case the
result is the right-hand side of (A.105). It should be noted that (A.107) may be used
to show Det A

Det A T .
The alternator may be used to express the fact that interchanging two rows or
two columns of a determinant changes the sign of the determinant,

¼

A 1m A 1n A 1p
A 2m A 2n A 2p
A 3m A 3n A 3p

A m1 A m2 A m3
A n1 e mnp Det A ¼ ¼ A n2 A n3 : (A.108)

A p1 A p2 A p3

Using the alternator again may combine these two representations:

A im A in A ip
A jm A jn A jp
A km A kn A kp

e ijk e mnp Det A ¼ : (A.109)

An Important Identity

In the special case when A ¼ 1 ( A ij ¼ d ij ), an important identity relating the
alternator to the Kronecker delta is obtained using (A.109):

d im d in d ip

e ijk e mnp ¼ d jm d jn d jp : (A.110)

d kn d kn d kp

The following special cases of (A.110) provide three more very useful relations
between the alternator and the Kronecker delta:

e mnk e ijk ¼ d im d jn d jm d in ; e mjk e ijk ¼ 2 d im ; e ijk e ijk ¼ 6 : (A.111)

The first of these relations is obtained by setting the indices p and k equal in
(A.110) and then expanding the determinant. The second is obtained from the first
by setting the indices n and j equal in the first. The third is obtained from the second
by setting the indices i and m equal in the second.

Example A.8.1

Derive the first of (A.111) from (A.110).

Solution: The first of (A.111) is obtained from (A.110) by setting the indices p and k
equal in (A.110) and then expanding the determinant: