Page 155 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

2.3 In this section we shall develop some of the fundamental properties of the
determinant function. Our work here will give us some further insight into the
PROPERTIES OF THE relationship between a square matrix and its determinant. One of the
DETERMINANT immediate consequences of this material will be the determinant test for the
FUNCTION invertibility of a matrix.

Basic Properties of Determinants

Suppose that A and B are matrices and is any scalar. We begin by considering possible relationships between ,
, and

Since a common factor of any row of a matrix can be moved through the det sign, and since each of the rows in has a
common factor of , we obtain

(1)

For example,

Unfortunately, no simple relationship exists among , , and . In particular, we emphasize that

will usually not be equal to . The following example illustrates this fact.

EXAMPLE 1
Consider

We have , , and ; thus

In spite of the negative tone of the preceding example, there is one important relationship concerning sums of determinants
that is often useful. To obtain it, consider two matrices that differ only in the second row:

We have