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

Step 2: Form a matrix with the minors calculated..

Matrix of Minors

=

M11 M12 M13 M21 M22 M23 M31 M32 M33

Step 3: Finding the cofactor from Minors:
Cofactor: A signed minor is called cofactor.
The cofactor of the element in the i^th row, j^th column is denoted by C_ij
C_ij = (-1)^i+j M_ij

Matrix of Cofactors	=	(-1)1+1M11 (-1)1+2M12 (-1)1+3M13 (-1)2+1M21 (-1)2+2M22 (-1)2+3M23 (-1)3+1M31 (-1)3+2M32 (-1)3+3M33
Matrix of Cofactors	=	C11 = 1 x M11 C12 = (-1) x M12 C13 = 1 x M13 C21 = (-1) x M21 C22 = 1 x M22 C23 = (-1) x M23 C31 = 1 x M31 C32 = (-1) x M32 C33 = 1 x M33

So,

C11 C12 C13 C21 C22 C23 C31 C32 C33

=

M11 -M12 M13 -M21 M22 -M23 M31 -M32 M33

Step 4: Calculate adjoint of matrix:
To calculate adjoint of matrix, just put the elements in rows to columns in the cofactor matrix. i.e convert the elements in first row to first column, second row to second column, third row to third column.

Adjoint of Matrix

=

C11 C21 C31 C12 C22 C32 C13 C23 C33