Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Please just 26 and 27
![1.6 EXERCISES
The matrices and vectors listed in Eq. (3) are used in
several of the exercises that follow.
31
121
-⠀ -⠀
A = 4 7
B = 7 4 3
26
601
2 40
--[!]
C =
6135
24 20
3 6
*=[28]
E=
3
D=[²4].
*=[¦}]
--[0) 46)
+-[3]
(3)
Exercises 1-25 refer to the matrices and vectors in
Eq. (3). In Exercises 1-6, perform the multiplications
to verify the given equality or nonequality.
1. (DE)F = D(EF)
3. DE # ED
2. (FE)D = F(ED)
4. EF # FE
6.3Fu = 7Fv
5. Fu = Fv
In Exercises 7-12, find the matrices.
7. AT
8. DT
10. ATC
11. (FV)
20. ||DV||
23. || Ful
9. ETF
12. (EF)v
In Exercises 13-25, calculate the scalars.
13. u7v
14. v¹ Fu
16. v¹ Fv
17. u u
19. u
22. ||u - v
25. ||(D-E)u||
26. Let A and B be (2x2) matrices. Prove or find a coun-
terexample for this statement: (AB)(A + B) =
A² - B².
15. v Dv
18. v'v
21. ||Au||
24. ||FV||
27. Let A and B be (2 x 2) matrices such that A² = AB
and A # O. Can we assert that, by cancellation,
A = B? Explain.
28. Let A and B be as in Exercise 27. Find the flaw in
the following proof that A = B.
Since A² = AB, A² - AB = O. Factoring
yields A(A - B) = O. Since A # O, it follows that
A - B = O. Therefore, A = B.
29. Two of the six matrices listed in Eq. (3) are symmet-
ric. Identify these matrices.
30. Find (2 x 2) matrices A and B such that A and
B are symmetric, but AB is not symmetric. [Hint:
(AB)¹ = B¹A¹ = BA.]
31. Let A and B be (n x n) symmetric matrices. Give
a necessary and sufficient condition for AB to be
symmetric. [Hint: Recall Exercise 30.]
32. Let G be the (2 x 2) matrix that follows, and con-
sider any vector x in R² where both entries are not
simultaneously zero:
--[G]-[R]
G = [ ¦ ¦ ] ; x = [
|x₁| + x₂ > 0.
Show that x¹Gx > 0. [Hint: Write x¹Gx as a sum
of squares.]
33. Repeat Exercise 32 using the matrix D in Eq. (3) in
place of G.
X2
34. For F in Eq. (3), show that xFx ≥ 0 for all x in
R². Classify those vectors x such that xFx=0.
If x and y are vectors in R", then the product x'y is of-
ten called an inner product. Similarly, the product xy
is often called an outer product. Exercises 35-40 con-
cern outer products; the matrices and vectors are given in
Eq. (3). In Exercises 35-40, form the outer products.
35. uv
38. u(Ev)
41. Let a and b be given by
36. u(Fu)
39. (Au) (Av)
37. v(Ev)
40. (Av) (Au)
-----
and b=
-[³]
a) Find x in R² that satisfies both x a = 6 and
x¹b = 2.
=
b) Find x in R² that satisfies both x²(a + b) = 12
and x¹a = 2.
42. Let A be a (2 x 2) matrix, and let B and C be given
by
13
*-[B]-[B]
and C=
a) If A¹ + B = C, what is A?
23
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Transcribed Image Text:1.6 EXERCISES
The matrices and vectors listed in Eq. (3) are used in
several of the exercises that follow.
31
121
-⠀ -⠀
A = 4 7
B = 7 4 3
26
601
2 40
--[!]
C =
6135
24 20
3 6
*=[28]
E=
3
D=[²4].
*=[¦}]
--[0) 46)
+-[3]
(3)
Exercises 1-25 refer to the matrices and vectors in
Eq. (3). In Exercises 1-6, perform the multiplications
to verify the given equality or nonequality.
1. (DE)F = D(EF)
3. DE # ED
2. (FE)D = F(ED)
4. EF # FE
6.3Fu = 7Fv
5. Fu = Fv
In Exercises 7-12, find the matrices.
7. AT
8. DT
10. ATC
11. (FV)
20. ||DV||
23. || Ful
9. ETF
12. (EF)v
In Exercises 13-25, calculate the scalars.
13. u7v
14. v¹ Fu
16. v¹ Fv
17. u u
19. u
22. ||u - v
25. ||(D-E)u||
26. Let A and B be (2x2) matrices. Prove or find a coun-
terexample for this statement: (AB)(A + B) =
A² - B².
15. v Dv
18. v'v
21. ||Au||
24. ||FV||
27. Let A and B be (2 x 2) matrices such that A² = AB
and A # O. Can we assert that, by cancellation,
A = B? Explain.
28. Let A and B be as in Exercise 27. Find the flaw in
the following proof that A = B.
Since A² = AB, A² - AB = O. Factoring
yields A(A - B) = O. Since A # O, it follows that
A - B = O. Therefore, A = B.
29. Two of the six matrices listed in Eq. (3) are symmet-
ric. Identify these matrices.
30. Find (2 x 2) matrices A and B such that A and
B are symmetric, but AB is not symmetric. [Hint:
(AB)¹ = B¹A¹ = BA.]
31. Let A and B be (n x n) symmetric matrices. Give
a necessary and sufficient condition for AB to be
symmetric. [Hint: Recall Exercise 30.]
32. Let G be the (2 x 2) matrix that follows, and con-
sider any vector x in R² where both entries are not
simultaneously zero:
--[G]-[R]
G = [ ¦ ¦ ] ; x = [
|x₁| + x₂ > 0.
Show that x¹Gx > 0. [Hint: Write x¹Gx as a sum
of squares.]
33. Repeat Exercise 32 using the matrix D in Eq. (3) in
place of G.
X2
34. For F in Eq. (3), show that xFx ≥ 0 for all x in
R². Classify those vectors x such that xFx=0.
If x and y are vectors in R", then the product x'y is of-
ten called an inner product. Similarly, the product xy
is often called an outer product. Exercises 35-40 con-
cern outer products; the matrices and vectors are given in
Eq. (3). In Exercises 35-40, form the outer products.
35. uv
38. u(Ev)
41. Let a and b be given by
36. u(Fu)
39. (Au) (Av)
37. v(Ev)
40. (Av) (Au)
-----
and b=
-[³]
a) Find x in R² that satisfies both x a = 6 and
x¹b = 2.
=
b) Find x in R² that satisfies both x²(a + b) = 12
and x¹a = 2.
42. Let A be a (2 x 2) matrix, and let B and C be given
by
13
*-[B]-[B]
and C=
a) If A¹ + B = C, what is A?
23
4 5
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