13. Solve for X in the equation, given A- and B- (a) W+ A- R (c) X- 3M + 28 =0 (d) 6X - 44 - 38 = (b) 24 - SR-Y
13. Solve for X in the equation, given A- and B- (a) W+ A- R (c) X- 3M + 28 =0 (d) 6X - 44 - 38 = (b) 24 - SR-Y
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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Done elementary_linear_algebra_8th_...
2.2 Exercises Se Calcchat.com tor worked out sclutons to od-numbered exercises.
Associativity of Matrix Multiplication In Exercises 23
and 24, find the matrix product ABC by (a) grouping
the factors as (AB)C, and (b) grouping the factors as
A(BC). Show that you obtain the same result from
both processes.
Evaluating an Expression In Exercises 1-6, evaluate
the expression.
-10
14
2.
23. A - ; -: c-; 1
24. A - [ -
B
I -5
-6
B
3
4. (15 -2 4 o1 + (14
0] +
6 -18
91)
-3
C-
5.
-5
Noncommutativity of Matrix Multiplication In
Exercises 25 and 26, show that AB and BA are not equal
for the given matrices.
3
4 +-9
13
6
Operations with Matrices In Exercises 7-12, perform
the operations, given a = 3, b = -4, and
25. A = 8-: 9
26. A-
B-
7. aA + bB
8. A +B
10. (a + biB
Equal Matrix Products In Exercises 27 and 28, show
that AC = BC, even though A + B.
9. abB)
11. (a - bA - B)
12. (ab)o
27. A - | -: c-[; |
B
13. Solve for X in the equation, given
3
28. A =0 5
2
4
-6-
3
-5 and B-
4. B
A=
-2
4
-1
[3 -2
o 0 0
C-0 0
4 -2
-3
4
4
(a) 3X + 24-R
(b) 24- SR- Y
(c) X - 34 + 2B = 0 (d) 6X - 44 - 3B = 0
3
14. Solve for X in the equation, given
-2 -1
I 0 and B = 2
3 -4
Zero Matrix Product In Exercises 29 and 30, show that
AB = 0, even though AO and B+ 0.
3
.
B--
-4 -1
29. A =
and B=
(a) X- 34 - 28
(b) 2X = 2A -R
(e) 2X + 3A = B
(d) 24 + 48 --2X
30. A =
and B=
Operations with Matrices In Exercises 15-22,
perform the operations, given e = -2 and
Operations with Matrices In Exercises 31-36,
perform the operations when
A -: : -: 1c - [; }
A =! 21
15. (BA)
16. (CB)
31. IA
32. AI
17. B(CA)
18. CIRCY
33. AI + A)
34. A+ IA
19. (B + C)A
20. B(C + 0)
35. A
36. 4
21. cB(C + C)
22. B(CA)
C Cl p
mal
tyco
king C
60
Chapter 2 Matrices
Writing In Exercises 37 and 38, explain why the
formula is not valid for matrices. Illustrate your
argument with examples.
Finding an nth Root of a Matrix In Exercises 53
and 54, find the nth root of the matrix B. An nth root of
a matrix R is a matrix A such that 4 = B.
[9
53. B-. n-2
37. (A + B)IA - B) - A-8
38. (A + B)(A + B) - A + 2AB + B
Finding the Transpose of a Matrix In Exercises 39
and 40, find the transpose of the matrix.
8
54. B-0 -1
27
ol -3
6 -7 19
40. D=-7 o 23
19 23 - 32
39. D--3
True or False? In Exercises 55 and 56, determine
whether each statement is true or false. If a statement
Finding the Transpose of a Product of Two Matrices
In Exercises 41-44, verify that (AB) = B'A".
is true, give a reason or cite an appropriate statement
from the text. f a statement is false, provide an example
that shows the statement is not true in all cases or cite an
-3
appropriate statement from the text.
55. (a) Matrix addition is commutative.
41. A=
and =
-1
(b) The transpose of the product of two matrices equals
the product of their transposes; that is, (AB) - A'B
(c) For any matrix C the matrix CC' is symmetric.
56. (a) Matrix multiplication is commutative.
42. A - 3 and B-
and B- 3!
(b) If the matrices A. B, and C satisfy AB - AC, then](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2Fe90b3609-4092-43e4-b8f6-cc6d860a7b17%2Fmm6hhjv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11:34
Done elementary_linear_algebra_8th_...
2.2 Exercises Se Calcchat.com tor worked out sclutons to od-numbered exercises.
Associativity of Matrix Multiplication In Exercises 23
and 24, find the matrix product ABC by (a) grouping
the factors as (AB)C, and (b) grouping the factors as
A(BC). Show that you obtain the same result from
both processes.
Evaluating an Expression In Exercises 1-6, evaluate
the expression.
-10
14
2.
23. A - ; -: c-; 1
24. A - [ -
B
I -5
-6
B
3
4. (15 -2 4 o1 + (14
0] +
6 -18
91)
-3
C-
5.
-5
Noncommutativity of Matrix Multiplication In
Exercises 25 and 26, show that AB and BA are not equal
for the given matrices.
3
4 +-9
13
6
Operations with Matrices In Exercises 7-12, perform
the operations, given a = 3, b = -4, and
25. A = 8-: 9
26. A-
B-
7. aA + bB
8. A +B
10. (a + biB
Equal Matrix Products In Exercises 27 and 28, show
that AC = BC, even though A + B.
9. abB)
11. (a - bA - B)
12. (ab)o
27. A - | -: c-[; |
B
13. Solve for X in the equation, given
3
28. A =0 5
2
4
-6-
3
-5 and B-
4. B
A=
-2
4
-1
[3 -2
o 0 0
C-0 0
4 -2
-3
4
4
(a) 3X + 24-R
(b) 24- SR- Y
(c) X - 34 + 2B = 0 (d) 6X - 44 - 3B = 0
3
14. Solve for X in the equation, given
-2 -1
I 0 and B = 2
3 -4
Zero Matrix Product In Exercises 29 and 30, show that
AB = 0, even though AO and B+ 0.
3
.
B--
-4 -1
29. A =
and B=
(a) X- 34 - 28
(b) 2X = 2A -R
(e) 2X + 3A = B
(d) 24 + 48 --2X
30. A =
and B=
Operations with Matrices In Exercises 15-22,
perform the operations, given e = -2 and
Operations with Matrices In Exercises 31-36,
perform the operations when
A -: : -: 1c - [; }
A =! 21
15. (BA)
16. (CB)
31. IA
32. AI
17. B(CA)
18. CIRCY
33. AI + A)
34. A+ IA
19. (B + C)A
20. B(C + 0)
35. A
36. 4
21. cB(C + C)
22. B(CA)
C Cl p
mal
tyco
king C
60
Chapter 2 Matrices
Writing In Exercises 37 and 38, explain why the
formula is not valid for matrices. Illustrate your
argument with examples.
Finding an nth Root of a Matrix In Exercises 53
and 54, find the nth root of the matrix B. An nth root of
a matrix R is a matrix A such that 4 = B.
[9
53. B-. n-2
37. (A + B)IA - B) - A-8
38. (A + B)(A + B) - A + 2AB + B
Finding the Transpose of a Matrix In Exercises 39
and 40, find the transpose of the matrix.
8
54. B-0 -1
27
ol -3
6 -7 19
40. D=-7 o 23
19 23 - 32
39. D--3
True or False? In Exercises 55 and 56, determine
whether each statement is true or false. If a statement
Finding the Transpose of a Product of Two Matrices
In Exercises 41-44, verify that (AB) = B'A".
is true, give a reason or cite an appropriate statement
from the text. f a statement is false, provide an example
that shows the statement is not true in all cases or cite an
-3
appropriate statement from the text.
55. (a) Matrix addition is commutative.
41. A=
and =
-1
(b) The transpose of the product of two matrices equals
the product of their transposes; that is, (AB) - A'B
(c) For any matrix C the matrix CC' is symmetric.
56. (a) Matrix multiplication is commutative.
42. A - 3 and B-
and B- 3!
(b) If the matrices A. B, and C satisfy AB - AC, then
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