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 |
|
= |
|
= |
a22xa33 - a32xa23 |
Minor of a12 = M12 |
= |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | a32 | a33 |
|
= |
|
= |
a21xa33 - a31xa23 |
Minor of a13 = M13 |
= |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | a32 | a33 |
|
= |
|
= |
a21xa32 - a31xa22 |
Minor of a21 = M21 |
= |
a11 | a12 | a13 |
a21 | a22 | a23 |
a31 | 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 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 M12 | C13 = 1 x M13 |
C21 = (-1) x M21 | C22 = 1 x M22 | C23 = (-1) x M23 |
C31 = 1 x M31 | C32 = (-1) xM32 | 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 |
|