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Minors and cofactors of a matrix

Definition.
Minor Mij to the element aij of the determinant of n order called the determinant of the (n - 1)-th order, derived from the original determinant by deleting the i-th row and j-th column.
Example 1.
Find the minors of matrix A
A = 
(571)
-410
203

Solution:

M11
5 7 1
-4 1 0
2 0 3
 = 
1 0
0 3
M11
1 0
0 3
 = 1·3 - 0·0 = 3 - 0 = 3
M12
-4 0
2 3
 = -4·3 - 0·2 = -12 -0 = -12
M13
-4 1
2 0
 = -4·0 - 1·2 = 0 - 2 = -2
M21
7 1
0 3
 = 7·3 - 1·0 = 21 - 0 = 21
M22
5 1
2 3
 = 5·3 - 1·2 = 15 - 2 = 13
M23
5 7
2 0
 = 5·0 - 7·2 = 0 - 14 = -14
M31
7 1
1 0
 = 7·0 - 1·1 = 0 - 1 = -1
M32
5 1
-4 0
 = 5·0 - 1·(-4) = 0 + 4 = 4
M33
5 7
-4 1
 = 5·1 - 7·(-4) = 5 + 28 = 33

Definition.
Cofactor Cij to element aij of determinant is number

Cij = (-1)i + j · Mij


Cofactors of matrix - properties

  • The sum of products of elements of row (column) of the determinant on the cofactors to the elements of this row (column) is equal to the determinant of the matrix:
    n
    Σaij·Aij = det(A)
    j = 1
  • The sum of products of elements of row (column) of the determinant on the cofactors to the elements of other row (column) is equal to zero:
    n
    Σakj·Aij = 0           (ik)
    j = 1
Example 2.
Find the cofactorsof matrix A
A = 
(571)
-410
203

Solution:

A11 = (-1)1 + 1·M11 = (-1)2·
1 0
0 3
 = 1·3 - 0·0 = 3 - 0 = 3
A12 = (-1)1 + 2·M12 = (-1)3·
-4 0
2 3
 = -(-4·3 - 0·2) = -(-12 -0) = 12
A13 = (-1)1 + 3·M13 = (-1)4·
-4 1
2 0
 = -4·0 - 1·2 = 0 - 2 = -2
A21 = (-1)2 + 1·M21 = (-1)3·
7 1
0 3
 = -(7·3 - 1·0) = -(21 - 0) = -21
A22 = (-1)2 + 2·M22 = (-1)4·
5 1
2 3
 = 5·3 - 1·2 = 15 - 2 = 13
A23 = (-1)2 + 3·M23 = (-1)5·
5 7
2 0
 = -(5·0 - 7·2) = -(0 - 14) = 14
A31 = (-1)3 + 1·M31 = (-1)4·
7 1
1 0
 = 7·0 - 1·1 = 0 - 1 = -1
A32 = (-1)3 + 2·M32 = (-1)5·
5 1
-4 0
 = -(5·0 - 1·(-4)) = -(0 + 4) = -4
A33 = (-1)3 + 3·M33 = (-1)6·
5 7
-4 1
 = 5·1 - 7·(-4) = 5 + 28 = 33

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