Minor of a Matrix





Definition of minor of a matrix:

Minor of a matrix may defined as follows, Let |A| = |[a ij]| be a determinant of order n. The minor of an arbitrary element aij is the determinant obtained by deleting the ith row and jth column in which the element aij stands. The minor of aij by Mij.

Co-factors:

The co-factor is a signed minor. The co-factor of aij is denoted by Aij and is defined as Aij = (-1) ^(i+j) Mij.

 
+ - +
- + -
+ - +
 

Example 1:

Find the minor and co-factor of the following matrix

 
3 4 1
0 -1 2
5 -2 6
 

Minor of 3

=
 
-1 2
-2 6
 

   = [-6-(-4)]

   = (-6+4)

   = -2

Minor of 4

=
 
0 2
5 6
 

   =  [0-10]

   =  (-10)

   = -10

Minor of 1

=
 
0 -1
5 -2
 

   =  [0-(-5)]

   =  [0+5]               minor of a matrix

   =  5

Minor of 0

=
 
4 1
-2 6
 

   = [24-(-2)]

   = [24+2]

   =  26

Minor of -1

=
 
3 1
5 6
 

   = [18-5]

   =  13

Minor of 2

=
 
3 4
5 -2
 

   = [-6-20]

   =  -26

Minor of 5

=
 
4 1
-1 2
 

   =  [8-(-1)]

   =  (8+1)

   =  9

Minor of -2

=
 
3 1
0 2
 

   =  [6-0]

   =  6

Minor of 6

=
 
3 4
0 -1
 

   =  [-3-0]

   =  -3

Minor matrix

 
-2 -10 5
26 13 -26
9 6 -3
 

Co-factor matrix

 
-2 10 5
-26 13 26
9 -6 -3
 

Example 2:

Find the minor and co-factor of the following matrix

 
3 1 -1
2 -2 0
1 2 -1
 

Minor of 3

=
 
-2 0
2 -1
 

   = [2-0]

   =  2

Minor of 1

=
 
2 0
1 -1
 

   = [-2-0]

   =  -2

Minor of -1

=
 
2 -2
1 2
 

   = [4-(-2)]

   = [4+2]

   =  6

Minor of 2

=
 
1 -1
2 -1
 

   = [-1-(-2)]

   = [-1+2]

   =  1

Minor of -2

=
 
3 -1
1 -1
 

   = [-3-(-1)]

   = [-3+1]

   =  -2

Minor of 0

=
 
3 1
1 2
 

   = [6-1]

   =  5

Minor of 1

=
 
1 -1
-2 0
 

   = [0-2]

   =  -2

Minor of 2

=
 
3 -1
2 0
 

   = [0-(-2)]

   =  2

Minor of -1

=
 
3 1
2 -2
 

   = [-6-2]

   =  -8

Minor matrix

 
2 -2 6
1 -2 5
-2 2 -8
 

Co-factor matrix

 
2 2 6
-1 -2 -5
-2 -2 -8
 

Here are the questions in minor of a matrix.












Questions



Solution


(1) Find the minor of the matrix

 
2 1 1
1 1 1
1 -1 2
 

Solution

(2) Find the minor of the matrix

 
1 2 3
1 1 1
2 3 4
 

Solution

(3) Find the minor of the matrix

 
6 2 3
3 1 1
10 3 4
 

Solution

(4) Find the minor of the matrix

 
1 1 -1
2 -3 4
3 -2 3
 

Solution

(5) Find the minor of the matrix

 
4 2 1
6 3 4
2 1 0
 

Solution


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Minor of a Matrix to Types of Matrices