Introduction to Electric Circuits

Cofactors

The cofactor of an element am,,
the minor of the element.

7.2 Matrices 145

in a square matrix is defined to be (-1)m+" times

Example 7.6

Find the cofactors of the elements in row 1 of the matrix

A ___ 1 3 4

2 3

1 -2

Solution

The cofactor of element all is found by removing the first row and the first

column and multiplying the determinant of the resulting 2 • 2 submatrix (i.e.

the minor of the element) by (--1) (1+1). Thus the cofactor of the element all is

(_1)~1+1)]2 3

1 -2

• (-1)2[(2 X -2)- (3 • 1)]- -7

Similarly, by removing the first row and the second column we find the cofactor

of a12. Therefore the cofactor of the element a12 is

(--1)( 1+2 )^2 3

1 -2

• (-1)3[(2 • -2) - (3 • 1)] : 7

We remove the first row and the third column to find the cofactor of the
element a13. Therefore the cofactor of the element a13 is

1)4[(2 • 1) - (2 • 1)1 = 0

Evaluation of A using cofactors

Determinants can be calculated in terms of cofactors using the rule

A- m=l E am~C,,,~ (7.1)

where x is the order of the square matrix, a,m is the element, and Cm~ is the

cofactor of the element.

Example 7.7

Evaluate the determinant of the matrix