Minor of a Matrix :
Let |A| = |[a ij]| be a determinant of order n.
The minor of an arbitrary element a_{ij} is the determinant obtained by deleting the i^{th} row and j^{th} column in which the element a_{ij} stands. The minor of a_{ij} by M_{ij}.
Cofactors :
The co factor is a signed minor. The cofactor of a_{ij} is denoted by A_{ij} and is defined as
A_{ij} = (-1)^{(i+j)} M_{ij}
Example 1 :
Find the minor and cofactor of the following matrix
Solution :
Minor of a_{11} (Ignore 1^{st} row and 1^{st} column)
Minor of a_{11} = -6+4 ==> -2
Minor of a_{12} = 0-10 ==> -10
Minor of a_{13} = 0+5 ==> 5
Minor of a_{21} = 24+2 ==> 26
Minor of a_{22} = 18-5 ==> 13
Minor of a_{23} = -6-20 ==> -26
Minor of a_{31} = 8+1 ==> 9
Minor of a_{32} = 6-0 ==> 6
Minor of a_{33} = -3-0 ==> -3
Example 2 :
Find the minor and cofactor of the following matrix
Solution :
Minor of a_{11} = 2-0 ==> 2
Minor of a_{12} = -2-0 ==> -2
Minor of a_{13} = 4+2 ==> 6
Minor of a_{21} = -1+2 ==> 1
Minor of a_{22} = -3+1 ==> -2
Minor of a_{23} = 6-1 ==> 5
Minor of a_{31} = 0-2 ==> -2
Minor of a_{32} = 0+2 ==> 2
Minor of a_{33} = -6-2 ==> -8
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