2.1 Matrix Operations 103 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?
2.1 Matrix Operations 103 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?
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
21
![vay in which
-performance
our definition
AB by rows.)
omputer. The
t processors,
orresponding
es of transposes
way to compute
e B is an n x p
entries in AB?
<7, what is the
matrix?
hat value(s) of
-[
5
3
3)
Com-
5
or rows of A
ght or on the
ix or the zero
Construct a 2 x 2 matrix B such that
AB is the zero matrix. Use two different nonzero columns
for B.
12. Let A
3
1
13. Let r..., p
be vectors in R", and let Q be an m × n matrix.
Write the matrix [Qr₁ Qrp] as a product of two matrices
(neither of which is an identity matrix).
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 q, 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 94 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].
16./a.
16./a. If A and B are 3 x 3 and B = [b₁ b₂ b3],then AB =
[Ab₁ + Ab₂+ Ab3].
15. a. If A and B are 2 × 2 with columns a₁, a2, and b₁,b2,
respectively, then AB = [a₁b₁a₂b₂].
b. Each column of AB is a linear combination of the columns
of B using weights from the corresponding column of A.
c. AB + AC = A (B+C)
nos yd.laupo sus AS bub
TAK
d. AT + BT = (A + B)¹
e. The transpose of a product of matrices equals the product
of their transposes in the same order.
b. The second row of AB is the second row of A multiplied
modliw.
Diw asnnw yllaue
on the right by B.
Co
c. (AB) C = (AC) B
d. (AB)T = AT BT
17. If A =
(x)
003 01
(2) bayons,
e. The transpose of a sum of matrices equals the sum of their
206 207000
transposes.
enou
=
08.14
-2
[13] and AB =[-!
-1 2 -1
6-9
-2
5
the first and second columns of B. = 1.dk
Exercises 15 and 16 concern arbitrary matrices A, B, and C for In Exercises 27 and 28, view vectors in R" as n x 1 matrices. For
of engul
which the indicated sums and products are defined. Mark each
d. Mark CachT
statement True or False. Justify each answer.
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 u²v and uvT will appear later in the text.
101 and
CATE determine
3]
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?
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?
20. Suppose the second column of B is all zeros. What can you
say about the second column of AB?
lawa less daw endog
gifion
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.
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.
2.1
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
oland C = D. [Hint: Think about the product CAD.]
Aib sid
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.
27. Let u =
vu.
-2
3
-4
Matrix Operations 103
and v
=
a
b
=
]
n
an (b₁j + c₁j) + ... + ain (bnj + Cnj) or Σaik (bkj + Czj)
k=1
Momo 30. Prove Theorem 2(d). [Hint: The (i, j)-entry in (rA)B is
(ra)bj++ (rain)bnj.]
C
=
Compute uv, v u, uv, and
28. If u and v are in R", how are uv and v' u related? How are
uv and vu 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
31. Show that Im A = A when A is an m x n matrix. You can
assume IX = x for all x in Rm.
rdk
33. Prove Theorem 3(d). [Hint: Consider the jth row of (AB)¹.]
18. Suppose the first two columns, b, and b2, of B are equal.
What can you say about the columns of AB (if AB is defined)? 34. Give a formula for (ABX), where x is a vector and A and B
Why? ans orb lle terit
tol 11,rodisgol enmulos bus are matrices of appropriate sizes.
Bus
m
32. Show that A In A when A is an m x n matrix. [Hint: Use
d) the (column) definition of AIn.]
IL
DETA
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). HT
a. A 5 x 6 matrix of zeros
b. A 3 x 5 matrix of ones](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F286f3f44-ea77-4623-ae9f-381dd0d9eeb1%2F721d8cfa-5990-45c1-b32d-83fc8d010cd3%2Flelwey7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:vay in which
-performance
our definition
AB by rows.)
omputer. The
t processors,
orresponding
es of transposes
way to compute
e B is an n x p
entries in AB?
<7, what is the
matrix?
hat value(s) of
-[
5
3
3)
Com-
5
or rows of A
ght or on the
ix or the zero
Construct a 2 x 2 matrix B such that
AB is the zero matrix. Use two different nonzero columns
for B.
12. Let A
3
1
13. Let r..., p
be vectors in R", and let Q be an m × n matrix.
Write the matrix [Qr₁ Qrp] as a product of two matrices
(neither of which is an identity matrix).
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 q, 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 94 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].
16./a.
16./a. If A and B are 3 x 3 and B = [b₁ b₂ b3],then AB =
[Ab₁ + Ab₂+ Ab3].
15. a. If A and B are 2 × 2 with columns a₁, a2, and b₁,b2,
respectively, then AB = [a₁b₁a₂b₂].
b. Each column of AB is a linear combination of the columns
of B using weights from the corresponding column of A.
c. AB + AC = A (B+C)
nos yd.laupo sus AS bub
TAK
d. AT + BT = (A + B)¹
e. The transpose of a product of matrices equals the product
of their transposes in the same order.
b. The second row of AB is the second row of A multiplied
modliw.
Diw asnnw yllaue
on the right by B.
Co
c. (AB) C = (AC) B
d. (AB)T = AT BT
17. If A =
(x)
003 01
(2) bayons,
e. The transpose of a sum of matrices equals the sum of their
206 207000
transposes.
enou
=
08.14
-2
[13] and AB =[-!
-1 2 -1
6-9
-2
5
the first and second columns of B. = 1.dk
Exercises 15 and 16 concern arbitrary matrices A, B, and C for In Exercises 27 and 28, view vectors in R" as n x 1 matrices. For
of engul
which the indicated sums and products are defined. Mark each
d. Mark CachT
statement True or False. Justify each answer.
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 u²v and uvT will appear later in the text.
101 and
CATE determine
3]
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?
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?
20. Suppose the second column of B is all zeros. What can you
say about the second column of AB?
lawa less daw endog
gifion
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.
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.
2.1
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
oland C = D. [Hint: Think about the product CAD.]
Aib sid
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.
27. Let u =
vu.
-2
3
-4
Matrix Operations 103
and v
=
a
b
=
]
n
an (b₁j + c₁j) + ... + ain (bnj + Cnj) or Σaik (bkj + Czj)
k=1
Momo 30. Prove Theorem 2(d). [Hint: The (i, j)-entry in (rA)B is
(ra)bj++ (rain)bnj.]
C
=
Compute uv, v u, uv, and
28. If u and v are in R", how are uv and v' u related? How are
uv and vu 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
31. Show that Im A = A when A is an m x n matrix. You can
assume IX = x for all x in Rm.
rdk
33. Prove Theorem 3(d). [Hint: Consider the jth row of (AB)¹.]
18. Suppose the first two columns, b, and b2, of B are equal.
What can you say about the columns of AB (if AB is defined)? 34. Give a formula for (ABX), where x is a vector and A and B
Why? ans orb lle terit
tol 11,rodisgol enmulos bus are matrices of appropriate sizes.
Bus
m
32. Show that A In A when A is an m x n matrix. [Hint: Use
d) the (column) definition of AIn.]
IL
DETA
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). HT
a. A 5 x 6 matrix of zeros
b. A 3 x 5 matrix of ones
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