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Minors and Cofactors

4.5 Minors and Cofactors

Let matrix be defined as the square matrix of order as shown below.
n
A
a 11 a 12 a 13 … a 1n

a 21 a 22 a 23 … a 2n
A = a 31 a 32 a 33 … a 3n (4.17)

……… … …
a n1 a n2 a n3 … a nn

If we remove the elements of its ith row, and jth column, the determinant of the remaining n – 1

A
square matrix is called the minor of determinant , and it is denoted as M .
ij

The signed minor –( 1 ) i + j M ij is called the cofactor of a ij and it is denoted as α ij .

Example 4.6

Given that

a 11 a 12 a 13
A = a 21 a 22 a 23 (4.18)
a 31 a 32 a 33

compute the minors M , M , M and the cofactors α , α and α .
11 12 13 11 12 13

Solution:

a a a a a a
M = 22 23 M = 21 23 M = 21 22
11
a 32 a 33 12 a 31 a 33 11 a 31 a 32
and

α 11 – (= 1 ) 1 + 1 M 11 = M 11 α 12 – ( = 1 ) 1 + 2 M 12 = – M 12 α 13 = M 13 – ( = 1 ) 1 + 3 M 13

The remaining minors
M 21 , M 22 , M 23 , M 31 , M 32 , M 33

and cofactors

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−13
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