Now, let A ij be the cofactors of the matrix elements a ij , then from Eq. 6.9, we
have,
A 11 ¼ ðÀ1Þ
1 þ 1 0 1
1 0
¼ À1; A 12 ¼ ðÀ1Þ
1 þ 2 1 1
1 0
¼ 1;
A 13 ¼ ðÀ1Þ
1 þ 3 1 0
1 1
¼ 1
A 21 ¼ ðÀ1Þ
2 þ 1 1 1
1 0
¼ 1; A 22 ¼ ðÀ1Þ
2 þ 2 0 1
1 0
¼ À1;
A 23 ¼ ðÀ1Þ
2 þ 3 0 1
1 1
¼ 1
A 31 ¼ ðÀ1Þ
3 þ 1 1 1
0 1
¼ 1; A 32 ¼ ðÀ1Þ
3 þ 2 0 1
1 1
¼ 1;
A 33 ¼ ðÀ1Þ
3 þ 3 0 1
1 0
¼ À1
So that the matrix of the cofactors is A ij ¼
À1
1
1
1 À1
1
1
1 À1
0
@
1
A
) AdjðAÞ ¼ A ji ¼
À1
1
1
1 À1
1
1
1 À1
0
@
1
A :
Hence,
A
À1
¼
AdjðAÞ
A
j j
¼
1
2
À1
1
1
1 À1
1
1
1 À1
0
@
1
A
Now,
A
À1 A ¼
1
2
À1
1
1
1 À1
1
1
1 À1
0
@
1
A
0 1 1
1 0 1
1 1 0
0
@
1
A ¼
1 0 0
0 1 0
0 0 1
0
@
1
A ¼ E ¼ I
Similarly,
AA
À1
¼
0 1 1
1 0 1
1 1 0
0
@
1
A
À
1
2
1
2
1
2
1
2 À
1
2
1
2
1
2
1
2 À
1
2
0
@
1
A ¼
1 0 0
0 1 0
0 0 1
0
@
1
A ¼ E ¼ I
234
6 Unit Cell Symmeteries and Their Representations