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Adjoint of a Matrix

Let A = [ a i j ] be a square matrix of order n . The adjoint of a matrix A is the transpose of the cofactor matrix of A . It is denoted by adj A . An adjoint matrix is also called an adjugate matrix.

Example:

Find the adjoint of the matrix.

A = [ 3       1 − 1 2 − 2       0 1       2 − 1 ]

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

Cofactor of       3 = A 11 =       | − 2 0 2 − 1 | = 2

Cofactor of       1 = A 12 = − | 2 0 1 − 1 | = 2

Cofactor of − 1 = A 13 =       | 2 − 2 1 2 | = 6

Cofactor of       2 = A 21 = − | 1 − 1 2 − 1 | = − 1

Cofactor of − 2 = A 22 =       | 3 − 1 1 − 1 | = − 2

Cofactor of       0 = A 23 = − | 3 1 1 2 | = − 5

Cofactor of       1 = A 31 =       | 1 − 1 − 2 0 | = − 2

Cofactor of       2 = A 32 = − | 3 − 1 2 0 | = − 2

Cofactor of − 1 = A 33 =       | 3 1 2 − 2 | = − 8

The cofactor matrix of A is [ A i j ] = [ 2 2 6 − 1 − 2 − 5 − 2 − 2 − 8 ]

Now find the transpose of A i j .

a d j   A = ( A i j ) T                           = [ 2 − 1 − 2 2 − 2 − 2 6 − 5 − 8 ]