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Adjoint of Matrix - Tutorial

  


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   =   
a11a12a13
a21a22a23
a31a32a33

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   =   
a11a12a13
a21a22a23
a31a32a33
   =   
a22a23
a32a33
   =    a22xa33 - a32xa23

Minor of a12 = M12   =   
a11a12a13
a21a22a23
a31a32a33
   =   
a21a23
a31a33
   =    a21xa33 - a31xa23

Minor of a13 = M13   =   
a11a12a13
a21a22a23
a31a32a33
   =   
a21a22
a31a32
   =    a21xa32 - a31xa22

Minor of a21 = M21   =   
a11a12a13
a21a22a23
a31a32a33
   =   
a12a13
a32a33
   =    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   =   
M11M12M13
M21M22M23
M31M32M33


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

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 M12C13 = 1 x M13
C21 = (-1) x M21C22 = 1 x M22C23 = (-1) x M23
C31 = 1 x M31C32 = (-1) xM32C33 = 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   =   
C11C21C31
C12C22C32
C13C23C33




Related Calculator:
>> Matrix Inverse & Adjoint Calculator.

Related Topics:
>> Find Inverse of matrix
>> Find determinant of A (|A|)
>> Calculate the inverse using the formula.