APPENDIX A Determinants and Cramer’s Rule 721
EXERCISES
1.Concept Check Decide whether each statement is trueor false. If a statement is false,
explain why.
(a)A matrix is an array of numbers, while a determinant is a single number.
(b)A square matrix has the same number of rows as columns.(c)The determinant is equal to(d)The value of is 0 for any replacements for xand y.2.Concept Check Which one of the following is the expression for the determinantA. B.
C. D.
Evaluate each determinant. See Example 1.3. 4. 5.
6. 7. 8.
Evaluate each determinant by expansion by minors about the first column. See Example 2.9. 10.
11. 12.
Evaluate each determinant by expansion by minors about any row or column. (Hint: The work
is easier if you choose a row or a column with 0 s.) See Example 3.13. 14. 15.
16. 17. 18.
19.Concept Check For the systemD=- 43 , Dx=- 43 , Dy= 0 , and Dz= 43. What is the solution set of the system?x+ 2 y+ 2 z=-1,2 x- y+ z= 1x+ 3 y- 6 z= 7(^3)
3
1
3
0
- 4
1
- 2
1
- 2
3 3
1
5
3
1
5
3
2
7
1
3 3
3
1
3
5
- 4
1
- 2
1
- 2
3
(^3)
0
4
2
0
0
- 1
3
- 2
3
3 3
4
1
1
4
- 1
0
2
- 2
2
3 3
3
1
0
- 1
5
2
2
- 2
0
3
(^3)
2
0
1
- 1
- 1
2
0
1
0
3 3
1
0
1
0
2
0
- 2
3
5
3
(^3)
2
1
5
- 3
2
3
- 5
2
- 1
3 3
- 1
- 3
2
2
- 2
- 1
4
- 3
5
3
`
8
0
- 3
0
` `
0
0
4
4
` `
- 5
1
- 1
0
`
`
1
7
- 2
0
` `
3
2
- 6
- 2
` `
- 2
- 1
5
4
`
- 3142 - 1 - 221 - 62 - 21 - 62 - 1 - 324
- 21 - 62 + 1 - 324 - 21 - 62 - 3142
`
- 2
4
- 3
- 6
`?
`
0
x0
y`
` ad+bc.a
cb
d