Page 129 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The cofactor of is
Similarly, the minor of entry is

The cofactor of is

Note that the cofactor and the minor of an element differ only in sign; that is, . A quick way to determine

whether to use + or is to use the fact that the sign relating and is in the th row and th column of the

“checkerboard” array

For example, , , , , and so on.

Strictly speaking, the determinant of a matrix is a number. However, it is common practice to “abuse” the terminology
slightly and use the term determinant to refer to the matrix whose determinant is being computed. Thus we might refer to

as a determinant and call 3 the entry in the first row and first column of the determinant.

Cofactor Expansions

The definition of a determinant in terms of minors and cofactors is

(1)

Equation 1 shows that the determinant of can be computed by multiplying the entries in the first row of by their
corresponding cofactors and adding the resulting products. More generally, we define the determinant of an matrix to
be

This method of evaluating is called cofactor expansion along the first row of .

EXAMPLE 2 Cofactor Expansion Along the First Row