Page 172 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

These expressions suggest the mnemonic devices given in Figure 2.4.2. The formula in part (a) of Example 7 is obtained from
Figure 2.4.2a by multiplying the entries on the rightward arrow and subtracting the product of the entries on the leftward arrow.
The formula in part (b) of Example 7 is obtained by recopying the first and second columns as shown in Figure 2.4.2b. The
determinant is then computed by summing the products on the rightward arrows and subtracting the products on the leftward
arrows.

Figure 2.4.2
Warning We emphasize that the methods shown in Figure 2.4.2 do not work for determinants of matrices or higher.

EXAMPLE 8 Evaluating Determinants
Evaluate the determinants of

Solution

Using the method of Figure 2.4.2a gives
Using the method of Figure 2.4.2b gives

The determinant of A may be written as

(1)

where indicates that the terms are to be summed over all permutations and the + or is selected in each term

according to whether the permutation is even or odd. This notation is useful when the combinatorial definition of a determinant

needs to be emphasized.

Remark Evaluating determinants directly from this definition leads to computational difficulties. Indeed, evaluating a