Page 275 - Advanced engineering mathematics

8.3 Evaluation of Determinants II 255

where G is the 2 × 2 matrix obtained by deleting row three and column two of F:

1153 1778
G = .
991 1510
This is 2 × 2 which we evaluate easily:

|G|= (1153)(1510) − (1778)(991) =−20,968.
Working back through the chain of determinants, we have

|A|=|B|=|C||D|
=−|E|=−|F|=|G|=−20,968.

SECTION 8.2 PROBLEMS

0
In each of Problems 1 through 10, use the method of this 1 1 −4

section to evaluate the determinant. In each problem there 7. 6 −3 2 2
are many different sequences of operations that can be used 1 −5 1 −2

to make the evaluation. 4 8 2 2

2 7 −1 0

−2 4 1 3 1 1 8
8.
1. 1 6 3 −2 0 3 1

7 0 4 4 8 −1 0

10
1 −6 2
2 −3 7
0 3 3 9
2. 14 1 9.

1
0 1 1 7
−13 −1

5
−2 6 8 8

−4 5 6 −7 16 2 4

3. −2 3 5 10. 1 0 0 5

2 −2 0 3 −4 4

6

6 1 1 −5

2 −5 8
11. Fill in the details of the following argument that
4. 4 3 8 |AB|=|A||B|.

13 0 −4

First, if AB is singular, show that at least one of A or B

17 −2 5 is singular, hence that the determinant of the product

5. 1 12 0 and the product of the determinants are both zero.

14 7

Thus, suppose that AB is nonsingular. Show that A
−7

−3 3 9 6 and B can be written as products of elementary matri-
ces, and then show that the determinant of a prod-
1 −2 15 6

6.
7 1 1 5 uct of elementary matrices equals the product of the

2 1 −1 3 determinants of these matrices.

8.3 Evaluation of Determinants II
In the preceding section, we evaluated determinants by using row and column operations to
produce rows and/or columns with all but one entry zero. In this section we exploit this idea from
a different perspective to write the determinant as a sum of numbers times smaller determinants.
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October 14, 2010 14:26 THM/NEIL Page-255 27410_08_ch08_p247-266