Page 130 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let . Evaluate by cofactor expansion along the first row of .

Solution

From 1,

If is a matrix, then its determinant is

(2)

(3)

By rearranging the terms in 3 in various ways, it is possible to obtain other formulas like 2. There should be no trouble
checking that all of the following are correct (see Exercise 28):

(4)

Note that in each equation, the entries and cofactors all come from the same row or column. These equations are called the

cofactor expansions of .

The results we have just given for matrices form a special case of the following general theorem, which we state
without proof.

THEOREM 2.1.1

Expansions by Cofactors

The determinant of an matrix can be computed by multiplying the entries in any row (or column) by their
cofactors and adding the resulting products; that is, for each and .

and