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.
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) 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.
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