A 11 ¼ ðÀ1Þ
1 þ 1 a 22 a 23
a 32 a 33
; A 12 ¼ ðÀ1Þ
1 þ 2 a 21 a 23
a 31 a 33
; A 13 ¼ ðÀ1Þ
1 þ 3 a 21 a 22
a 31 a 32
A 21 ¼ ðÀ1Þ
2 þ 1 a 12 a 13
a 32 a 33
; A 22 ¼ ðÀ1Þ
2 þ 2 a 11 a 13
a 31 a 33
; A 23 ¼ ðÀ1Þ
2 þ 3 a 11 a 12
a 31 a 32
A 31 ¼ ðÀ1Þ
3 þ 1 a 12 a 13
a 22 a 23
; A 32 ¼ ðÀ1Þ
3 þ 2 a 11 a 13
a 21 a 23
; A 33 ¼ ðÀ1Þ
3 þ 3 a 11 a 12
a 21 a 22
The matrix of the cofactors, A ij is given by
A ij ¼
A 11 A 12 A 13
A 21 A 22 A 23
A 31 A 32 A 33
0
@
1
A
Transpose of the matrix A ij is the Adj(A)
AdjðAÞ ¼ A ji ¼
A 11 A 21 A 31
A 12 A 22 A 32
A 13 A 23 A 33
0
@
1
A
Hence, the inverse is obtained as
A
À1
¼
AdjðAÞ
A
j j
¼
1
A
j j
A 11 A 21 A 31
A 12 A 22 A 32
A 13 A 23 A 33
0
@
1
A
Example 14 Find the inverse of the matrix A ¼
0 1 1
1 0 1
1 1 0
0
@
1
A and verify that
A
−1 A = AA
−1 = EE.
Solution: Given: the matrix A ¼
0 1 1
1 0 1
1 1 0
0
@
1
A
Now, the determinant of the given matrix is
A
j j ¼
0 1 1
1 0 1
1 1 0
¼ 0ð0 À 1Þ À 1 0 À 1
ð
Þþ1 1 À 0
ð
Þ¼1 þ 1 ¼ 2
⟹ A
−1 exists.
6.2 Matrix Representation of Symmetry Operations
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