50 Chapter 2 Matrices 61. Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AR - RA Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. () Begin your proof by letting A = [a,) and B = [b,] be two diagonal nxn matrices. (ii) The tth entry of the product AB is (iii) Evaluate the entries e, for the two cases i j and i-j. (iv) Repeat this analysis for the product BA.
50 Chapter 2 Matrices 61. Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AR - RA Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. () Begin your proof by letting A = [a,) and B = [b,] be two diagonal nxn matrices. (ii) The tth entry of the product AB is (iii) Evaluate the entries e, for the two cases i j and i-j. (iv) Repeat this analysis for the product BA.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Topic Video
Question
100%
#61
![| sin cos
1:40
Done elementary_linear_algebra_8th_...
50
Chapter 2 Matrices
61. Guided Proof Prove that if A and B are diagonal
72. Show that no 2 x 2 matrices A and R exist that sa
matrices (of the same size), then AR = RA.
the matrix equation
AB - BA =
Getting Started: To prove that the matrices AB and BA
are equal, you need to show that their corresponding
entries are equal.
(i) Begin your proof by letting A = [a,] and B = [b,]
be two diagonal n x n matrices.
(ii) The ijth entry of the product AB is
73. Exploration Let i = /-1 and let
-: and B=
(a) Find A. A', and A*. (Note: A? =
A' = AAA - A'A, and so on.) Identify any similam
with f, P, and i.
(iii) Evaluate the entries cy for the two cases i + j and
i = j.
(b) Find and identify B.
74. Guided Proof Prove that if the product AB is a sq
matrix, then the product RA is defined.
Getting Started: To prove that the product B
defined, you need to show that the number of colu
of R equals the number of rows of A.
(iv) Repeat this analysis for the product BA.
62. Writing Let A and R be 3 x 3 matrices, where A is
diagonal.
(a) Describe the product AB. Illustrate your answer
with examples.
(i) Begin your proof by noting that the numbe
columns of A equals the number of rows of R.
(b) Describe the product BA. Illustrate your answer
with examples.
(ii) Then assume that A has size m x n and B has
(c) How do the results in parts (a) and (b) change when
the diagonal entries of A are all equal?
nx p.
Use the hypothesis that the product AB
square matrix.
Trace of a Matrix In Exercises 63-66, find the
75. Proof Prove that if both products AB and BA
defined, then AR and RA are square matrices.
trace of the matrix. The trace of an n xn matrix A
is the sum of the main diagonal entries. That is,
Tr(A) = a, + an + + a
[1
63. |о -2
[3
[1
76. Let A and B be matrices such that the product A
defined. Show that if A has two identical rows, then
[1
64. 0
2
3
corresponding two rows of AB are also identical.
4
77. Let A and R be nxn matrices. Show that if the
row of A has all zero entries, then the ith row of
will have all zero entries. Give an example using 2
3
[1
4
2
4
3
2
-1
2
0 6
matrices to show that the converse is not true.
65.
4
6.
3
6 2
0 5
[2
1 1 -3
78. CAPSTONE Let matrices A and R be
sizes 3 x 2 and 2 x 2, respectively. Answer ea
question and explain your answers.
67. Proof Prove that each statement is true when A and B
are square matrices of order n and c is a scalar.
(a) Tr(A + B) = Tr(A) + Tr(B)
(a) Is it possible that A = B?
(b) Is A + R defined?
(b) Tr(cA) = cTr(A)
(c) Is AB defined? If so, is it possible that AB = BA
68. Proof Prove that if A and B are square matrices of
order n, then Tr(AB)- Trí(BA).
69. Find conditions on w, x, y, and z such that AB = BA for
79. Agriculture A fruit grower raises two crops, ap
and peaches. The grower ships each of these crop
three different outlets. In the matrix
125 100 75]
the matrices below
and B =
100 175 125
70. Verify AB = BA for the matrices below.
a, represents the number of units of crop i that
grower ships to outlet j. The matrix
B = [53.50 $6.00]
cos a -sin a
cos B - sin B
A =
and B=
sin a
cos a
sin B
cos B]
71. Show that the matrix equation has no solution.
represents the profit per unit. Find the product RA
state what each entry of the matrix represents.
Cde 27 Conr kemine AI Re Reened Mehe id d rd in wepn Derlec e, w pe me he ed e h
Ed a ded s ly ling pi Congae Leng ene y
2.1 Exercise
80. Manufacturing A corporation has three factories,
each of which manufactures acoustic guitars and
electric guitars. In the matrix
Block Multiplication In Exercises 83 and 84.
block multiplication of matrices A and R.
A and B are each partitioned into four submat
[Bu B12
-B.. B.
A =| 70 50 25
701
35 100
A = [A Au and B=
A..
A..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F28575a1e-ba0e-484d-8458-6e509e7534df%2Fe6h6218_processed.jpeg&w=3840&q=75)
Transcribed Image Text:| sin cos
1:40
Done elementary_linear_algebra_8th_...
50
Chapter 2 Matrices
61. Guided Proof Prove that if A and B are diagonal
72. Show that no 2 x 2 matrices A and R exist that sa
matrices (of the same size), then AR = RA.
the matrix equation
AB - BA =
Getting Started: To prove that the matrices AB and BA
are equal, you need to show that their corresponding
entries are equal.
(i) Begin your proof by letting A = [a,] and B = [b,]
be two diagonal n x n matrices.
(ii) The ijth entry of the product AB is
73. Exploration Let i = /-1 and let
-: and B=
(a) Find A. A', and A*. (Note: A? =
A' = AAA - A'A, and so on.) Identify any similam
with f, P, and i.
(iii) Evaluate the entries cy for the two cases i + j and
i = j.
(b) Find and identify B.
74. Guided Proof Prove that if the product AB is a sq
matrix, then the product RA is defined.
Getting Started: To prove that the product B
defined, you need to show that the number of colu
of R equals the number of rows of A.
(iv) Repeat this analysis for the product BA.
62. Writing Let A and R be 3 x 3 matrices, where A is
diagonal.
(a) Describe the product AB. Illustrate your answer
with examples.
(i) Begin your proof by noting that the numbe
columns of A equals the number of rows of R.
(b) Describe the product BA. Illustrate your answer
with examples.
(ii) Then assume that A has size m x n and B has
(c) How do the results in parts (a) and (b) change when
the diagonal entries of A are all equal?
nx p.
Use the hypothesis that the product AB
square matrix.
Trace of a Matrix In Exercises 63-66, find the
75. Proof Prove that if both products AB and BA
defined, then AR and RA are square matrices.
trace of the matrix. The trace of an n xn matrix A
is the sum of the main diagonal entries. That is,
Tr(A) = a, + an + + a
[1
63. |о -2
[3
[1
76. Let A and B be matrices such that the product A
defined. Show that if A has two identical rows, then
[1
64. 0
2
3
corresponding two rows of AB are also identical.
4
77. Let A and R be nxn matrices. Show that if the
row of A has all zero entries, then the ith row of
will have all zero entries. Give an example using 2
3
[1
4
2
4
3
2
-1
2
0 6
matrices to show that the converse is not true.
65.
4
6.
3
6 2
0 5
[2
1 1 -3
78. CAPSTONE Let matrices A and R be
sizes 3 x 2 and 2 x 2, respectively. Answer ea
question and explain your answers.
67. Proof Prove that each statement is true when A and B
are square matrices of order n and c is a scalar.
(a) Tr(A + B) = Tr(A) + Tr(B)
(a) Is it possible that A = B?
(b) Is A + R defined?
(b) Tr(cA) = cTr(A)
(c) Is AB defined? If so, is it possible that AB = BA
68. Proof Prove that if A and B are square matrices of
order n, then Tr(AB)- Trí(BA).
69. Find conditions on w, x, y, and z such that AB = BA for
79. Agriculture A fruit grower raises two crops, ap
and peaches. The grower ships each of these crop
three different outlets. In the matrix
125 100 75]
the matrices below
and B =
100 175 125
70. Verify AB = BA for the matrices below.
a, represents the number of units of crop i that
grower ships to outlet j. The matrix
B = [53.50 $6.00]
cos a -sin a
cos B - sin B
A =
and B=
sin a
cos a
sin B
cos B]
71. Show that the matrix equation has no solution.
represents the profit per unit. Find the product RA
state what each entry of the matrix represents.
Cde 27 Conr kemine AI Re Reened Mehe id d rd in wepn Derlec e, w pe me he ed e h
Ed a ded s ly ling pi Congae Leng ene y
2.1 Exercise
80. Manufacturing A corporation has three factories,
each of which manufactures acoustic guitars and
electric guitars. In the matrix
Block Multiplication In Exercises 83 and 84.
block multiplication of matrices A and R.
A and B are each partitioned into four submat
[Bu B12
-B.. B.
A =| 70 50 25
701
35 100
A = [A Au and B=
A..
A..
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
