5 1  Adjoint of a matrix
Definition 3 The adjoint of a square matrix A = [aij]n × n is defined as the transpose of
the matrix [Aij]n × n, where Aij is the cofactor of the element aij Adjoint of the matrix A
is denoted by adj A Let
11
12
13
21
22
23
31
32
33
A =
a
a
a
a
a
a
a
a
a










Rationalised 2023-24
 88
MATHEMATICS
Then
11
12
13
21
22
23
31
32
33
A
A
A
A =Transposeof A
A
A
A
A
A
adj










11
21
31
12
22
32
13
23
33
A
A
A
= A
A
A
A
A
A










Example 12 
2
3
Find 
 A for A = 1
4
adj






Solution We have  A11 = 4, A12 = –1, A21 = –3, A22 = 2
Hence
adj A =
11
21
12
22
A
A
4
–3
 =
A
A
–1
2












Remark For a square matrix of order 2, given by
A =
11
12
21
22
a
a
a
a






The adj A can also be obtained by interchanging a11 and a22 and by changing signs
of a12 and a21, i