I am explaining by taking example 3x3 matrix

 

Adjoint of Matrix :
      Adjoint or Adjugate Matrix of a square matrix is the transpose of the matrix formed by the cofactors of elements of determinant |A|.

To calculate adjoint of matrix we have to follow the procedure
a) Calculate Minor for each element of the matrix.
b) Form Cofactor matrix from the minors calculated.
c) Form Adjoint from cofactor matrix.

For an example we will use a matrix A

Matrix A

=

a11 a12 a13 a21 a22 a23 a31 a32 a33

Step 1: Calculate Minor for each element.
       To calculate the minor for an element we have to use the elements that do not fall in the same row and column of the minor element.

Minor of a11 = M11

=

a11 a12 a13 a21 a22 a23 a31 a32 a33

=

a22 a23 a32 a33

=

a22xa33 - a32xa23

Minor of a12 = M12

=

a11 a12 a13 a21 a22 a23 a31 a32 a33

=

a21 a23 a31 a33

=

a21xa33 - a31xa23

Minor of a13 = M13

=

a11 a12 a13 a21 a22 a23 a31 a32 a33

=

a21 a22 a31 a32

=

a21xa32 - a31xa22

Minor of a21 = M21

=

a11 a12 a13 a21 a22 a23 a31 a32 a33

=

a12 a13 a32 a33

=

a12xa33 - a32xa13

Similarly M22 = a11xa33 - a31xa13         M23 = a11xa32 - a31xa12 M31 = a12xa23 - a22xa13         M32 = a11xa23 - a21xa13 M33 = a11xa22 - a21xa12

 

Example:	Find the adjoint of the following matrix:
Solution:	First find the cofactor of each element. As a result the cofactor matrix of A is Finally the adjoint of A is the transpose of the cofactor matrix: