Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 367

and the representation for the multiplication of A times a ,

2 3 2 3 2 3

A 11 A 12 A 13
A 21 A 22 A 23
A 31 A 32 A 33

a 1
a 2
a 3

A 11 a 1 þ A 12 a 2 þ A 13 a 3

4 5 4 5 ¼ 4 5 ;

A a ¼ A 21 a 1 þ A 22 a 2 þ A 23 a 3

A 31 a 1 þ A 32 a 2 þ A 33 a 3

then

A a ðA b A cÞ

2
4

3
5:

A 11 a 1 þ A 12 a 2 þ A 13 a 3 A 21 a 1 þ A 22 a 2 þ A 23 a 3 A 31 a 1 þ A 32 a 2 þ A 33 a 3
A 11 b 1 þ A 12 b 2 þ A 13 b 3 A 21 b 1 þ A 22 b 2 þ A 23 b 3 A 31 b 1 þ A 32 b 2 þ A 33 b 3
A 11 c 1 þ

¼ Det

A 12 c 2 þ A 13 c 3 A 21 c 1 þ A 22 c 2 þ A 23 c 3 A 31 c 1 þ A 32 c 2 þ A 33 c 3

Det A Det B , it follows that the
previous determinant may be written as a product of determinants,

Recalling from Example A.8.1 that Det( A B ) ¼

2 3 2 3

A 11 A 12 A 13
A 21 A 22 A 23
A 31 A 32 A 33

a 1 b 1 c 1

4 5 Det 4 5 ¼

Det a 2 b 2 c 2 a ð b c Þ DetA ;

a 3 b 3 c 3

which is the desired result. In the last step the fact that the determinant of the
transpose of a matrix is equal to the determinant of the matrix, Det A Det A T , was ¼

employed.

□

Example A.8.6

Prove vector identity ( A b A c )· A ¼ ( b c ) Det A . where A is a 3 by 3 matrix

and b and c are vectors.
Solution: Recall the result of Example A8.5, namely that A a ·( A b A c ) ¼ a ·( b

c )Det A ,andlet a ¼ e 1 ,then e 2 and then e 3 to obtain the following three scalar

equations:

A 11 w 1 þ A 21 w 2 þ A 31 w 3 ¼ q 1 Det A

A 12 w 1 þ A 22 w 2 þ A 32 w 3 ¼ q 2 Det A

A 13 w 1 þ A 23 w 2 þ A 33 w 3 ¼ q 3 Det A

where w ¼ ( A b A c ), q ¼ ( b c ). These three equations many be recast in the

matrix notation,

2
4

3
5

2
4

3
5 ¼

2
4

3
5 Det A

A 11 A 21 A 31
A 12 A 22 A 32
A 13 A 23 A 33

w 1
w 2
w 3

q 1
q 2
q 3

;