have more rows than columns. 25. Suppose A is an m x n matrix and there exist n x m matrices C and D such that CA = I, and AD = Im. Prove that m = n and C = D. [Hint: Think about the product CAD.] 26 Suppose 4 is R3 E.
have more rows than columns. 25. Suppose A is an m x n matrix and there exist n x m matrices C and D such that CA = I, and AD = Im. Prove that m = n and C = D. [Hint: Think about the product CAD.] 26 Suppose 4 is R3 E.
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
Question
25
![ance
WS.)
The
ors,
P
B
-6
[3]
2
12. Let A =
Construct a 2 x 2 matrix B such that
for B.
AB is the zero matrix. Use two different nonzero columns
13. Let r₁,..., rp be vectors in R", and let Q be an m x n matrix.
Write the matrix [ Qr₁ Qr, ] as a product of two matrices
(neither of which is an identity matrix).
22110
14. Let U be the 3 x 2 cost matrix described in Example 6 of
Section 1.8. The first column of U lists the costs per dollar of
output for manufacturing product B, and the second column
lists the costs per dollar of output for product C. (The costs
are categorized as materials, labor, and overhead.) Let q₁ be
a vector in R2 that lists the output (measured in dollars) of
products B and C manufactured during the first quarter of
the year, and let 92, 93, and q4 be the analogous vectors
that list the amounts of products B and C manufactured in
the second, third, and fourth quarters, respectively. Give an
economic description of the data in the matrix UQ, where
Q = [91 92 93 94].
[q₁
Exercises 15 and 16 concern arbitrary matrices A, B, and C for
which the indicated sums and products are defined. Mark each
statement True or False. Justify each answer.
SNELUCH
10
***
15. a. If A and B are 2 × 2 with columns a₁, a2, and b₁,b₂,
respectively, then AB = [a₁b₁ a₂b₂].
b. Each column of AB is a linear combination of the columns
of B using weights from the corresponding column of A.
yd jeupe sis
c. AB + AC = A (B+C)
d. AT + BT = (A + B)T 110
ET
e. The transpose of a product of matrices equals the product
of their transposes in the same order.
16./a.
75
16./a. If A and B are 3 x 3 and B = [b₁ b2 b3],then AB =
[Ab₁ + Ab₂ + Ab3].
fenoodT
17. If A =
b. The second row of AB is the second row of A multiplied
on the right by B.
odir nany llegen
070
c. (AB) C = (AC) B
d. (AB) = AT BT
ose of as
00012
e. The transpose of a sum of matrices equals the sum of their
transposes.
-1-2
the first and second columns of B.
-2
-2 5
sing
-1
- [
6
and AB =
2 -1
-9
1, determine
3
18. Suppose the first two columns, b, and b2, of B are equal.
sm What can you say about the columns of AB (if AB is defined)?
Why?
ogol analos
on lis jadi evollot
19. Suppose the third column of B is the sum of the first two
columns. What can you say about the third column of AB?
Why?
20. Suppose the second column of B is all zeros. What can you
say about the second column of AB?
s bazinodn
las dire
21. Suppose the last column of AB is entirely zero but B itself
has no column of zeros. What can you say about the columns
of A?
22. Show that if the columns of B are linearly dependent, then
so are the columns of AB.
23. Suppose CA = I, (the n x n identity matrix). Show that the
equation Ax = 0 has only the trivial solution. Explain why
A cannot have more columns than rows.
2.1 Matrix Operations 103
24. Suppose AD = Im (the mxm identity matrix). Show that
for any b in R", the equation Ax = b has a solution. [Hint:
Think about the equation ADb = b.] Explain why A cannot
have more rows than columns.
25. Suppose A is an m x n matrix and there exist n x m matrices
C and D such that CA = In and AD = Im. Prove that m = n
and C = D. [Hint: Think about the product CAD.]
ol
26. Suppose A is a 3 x n matrix whose columns span R3. Explain
how to construct an n x 3 matrix D such that AD = I3.
ucu
In Exercises 27 and 28, view vectors in R" as n x 1 matrices. For
u and v in R", the matrix product u v is a 1 x 1 matrix, called the
scalar product, or inner product, of u and v. It is usually written
as a single real number without brackets. The matrix product uv
is an n x n matrix, called the outer product of u and v. The
products u v and uv will appear later in the text.
27. Let u =
228.
-2
[]
-4
and y =
a
b
C
. Compute uv, v u, uv, and
vUT.
If u and v are in R", how are uv and v u related? How are
uv and vu¹ related?
29. Prove Theorem 2(b) and 2(c). Use the row-column rule. The
(i, j)-entry in A(B+C) can be written as
n
...
A¡1 (b₁j + C₁j) + ··· + Ain (bnj + Cnj) or Σaik (bkj + Ckj)
k=1
30. Prove Theorem 2(d). [Hint: The (i, j)-entry in (rA)B is
(ra)b₁j+...+ (rain)bnj.]
31. Show that Im A = A when A is an m x n matrix. You can
assume ImX = x for all x in Rm.
32. Show that AI = A when A is an m x n matrix. [Hint: Use
dA] the (column) definition of AIn.]
33. Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
34. Give a formula for (ABX)T, where x is a vector and A and B
are matrices of appropriate sizes.
35. [M] Read the documentation for your matrix program, and
write the commands that will produce the following matrices
(without keying in each entry of the matrix).
a. A 5 x 6 matrix of zeros
b. A3 x 5 matrix of ones](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12f186d1-b037-4d7e-ab2f-58fa2244e958%2F301b9f05-b5d6-46ff-ba97-da0902d6dfaa%2Fozrki1n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ance
WS.)
The
ors,
P
B
-6
[3]
2
12. Let A =
Construct a 2 x 2 matrix B such that
for B.
AB is the zero matrix. Use two different nonzero columns
13. Let r₁,..., rp be vectors in R", and let Q be an m x n matrix.
Write the matrix [ Qr₁ Qr, ] as a product of two matrices
(neither of which is an identity matrix).
22110
14. Let U be the 3 x 2 cost matrix described in Example 6 of
Section 1.8. The first column of U lists the costs per dollar of
output for manufacturing product B, and the second column
lists the costs per dollar of output for product C. (The costs
are categorized as materials, labor, and overhead.) Let q₁ be
a vector in R2 that lists the output (measured in dollars) of
products B and C manufactured during the first quarter of
the year, and let 92, 93, and q4 be the analogous vectors
that list the amounts of products B and C manufactured in
the second, third, and fourth quarters, respectively. Give an
economic description of the data in the matrix UQ, where
Q = [91 92 93 94].
[q₁
Exercises 15 and 16 concern arbitrary matrices A, B, and C for
which the indicated sums and products are defined. Mark each
statement True or False. Justify each answer.
SNELUCH
10
***
15. a. If A and B are 2 × 2 with columns a₁, a2, and b₁,b₂,
respectively, then AB = [a₁b₁ a₂b₂].
b. Each column of AB is a linear combination of the columns
of B using weights from the corresponding column of A.
yd jeupe sis
c. AB + AC = A (B+C)
d. AT + BT = (A + B)T 110
ET
e. The transpose of a product of matrices equals the product
of their transposes in the same order.
16./a.
75
16./a. If A and B are 3 x 3 and B = [b₁ b2 b3],then AB =
[Ab₁ + Ab₂ + Ab3].
fenoodT
17. If A =
b. The second row of AB is the second row of A multiplied
on the right by B.
odir nany llegen
070
c. (AB) C = (AC) B
d. (AB) = AT BT
ose of as
00012
e. The transpose of a sum of matrices equals the sum of their
transposes.
-1-2
the first and second columns of B.
-2
-2 5
sing
-1
- [
6
and AB =
2 -1
-9
1, determine
3
18. Suppose the first two columns, b, and b2, of B are equal.
sm What can you say about the columns of AB (if AB is defined)?
Why?
ogol analos
on lis jadi evollot
19. Suppose the third column of B is the sum of the first two
columns. What can you say about the third column of AB?
Why?
20. Suppose the second column of B is all zeros. What can you
say about the second column of AB?
s bazinodn
las dire
21. Suppose the last column of AB is entirely zero but B itself
has no column of zeros. What can you say about the columns
of A?
22. Show that if the columns of B are linearly dependent, then
so are the columns of AB.
23. Suppose CA = I, (the n x n identity matrix). Show that the
equation Ax = 0 has only the trivial solution. Explain why
A cannot have more columns than rows.
2.1 Matrix Operations 103
24. Suppose AD = Im (the mxm identity matrix). Show that
for any b in R", the equation Ax = b has a solution. [Hint:
Think about the equation ADb = b.] Explain why A cannot
have more rows than columns.
25. Suppose A is an m x n matrix and there exist n x m matrices
C and D such that CA = In and AD = Im. Prove that m = n
and C = D. [Hint: Think about the product CAD.]
ol
26. Suppose A is a 3 x n matrix whose columns span R3. Explain
how to construct an n x 3 matrix D such that AD = I3.
ucu
In Exercises 27 and 28, view vectors in R" as n x 1 matrices. For
u and v in R", the matrix product u v is a 1 x 1 matrix, called the
scalar product, or inner product, of u and v. It is usually written
as a single real number without brackets. The matrix product uv
is an n x n matrix, called the outer product of u and v. The
products u v and uv will appear later in the text.
27. Let u =
228.
-2
[]
-4
and y =
a
b
C
. Compute uv, v u, uv, and
vUT.
If u and v are in R", how are uv and v u related? How are
uv and vu¹ related?
29. Prove Theorem 2(b) and 2(c). Use the row-column rule. The
(i, j)-entry in A(B+C) can be written as
n
...
A¡1 (b₁j + C₁j) + ··· + Ain (bnj + Cnj) or Σaik (bkj + Ckj)
k=1
30. Prove Theorem 2(d). [Hint: The (i, j)-entry in (rA)B is
(ra)b₁j+...+ (rain)bnj.]
31. Show that Im A = A when A is an m x n matrix. You can
assume ImX = x for all x in Rm.
32. Show that AI = A when A is an m x n matrix. [Hint: Use
dA] the (column) definition of AIn.]
33. Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
34. Give a formula for (ABX)T, where x is a vector and A and B
are matrices of appropriate sizes.
35. [M] Read the documentation for your matrix program, and
write the commands that will produce the following matrices
(without keying in each entry of the matrix).
a. A 5 x 6 matrix of zeros
b. A3 x 5 matrix of ones
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