HOW TO FIND MINORS AND COFACTORS OF A MATRIX

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 a₁₁ (Ignore 1^st row and 1^st column)

Minor of a₁₁  =  -6+4  ==>  -2

Minor of a₁₂  =  0-10  ==>  -10

Minor of a₁₃  =  0+5  ==>  5

Minor of a₂₁  =  24+2  ==>  26

Minor of a₂₂  =  18-5  ==>  13

Minor of a₂₃  =  -6-20  ==>  -26

Minor of a₃₁  =  8+1  ==>  9

Minor of a₃₂  =  6-0  ==>  6

Minor of a₃₃  =  -3-0  ==>  -3 

Example 2 :

Find the minor and cofactor of the following matrix

Solution :

Minor of a₁₁  =  2-0  ==>  2

Minor of a₁₂  =  -2-0  ==>  -2

Minor of a₁₃  =  4+2  ==>  6

Minor of a₂₁  =  -1+2  ==>  1

Minor of a₂₂  =  -3+1  ==>  -2

Minor of a₂₃  =  6-1  ==>  5

Minor of a₃₁  =  0-2  ==>  -2

Minor of a₃₂  =  0+2  ==>  2

Minor of a₃₃  =  -6-2  ==>  -8 

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