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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 = 5 7 1 -4 1 0 2 0 3

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 Σ aij·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 Σ akj·Aij = 0           (i ≠ k) j = 1
Example 2.
Find the cofactorsof matrix A
A = 5 7 1 -4 1 0 2 0 3

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