Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are defined. a. A(BC) = (AB)C b. A(B+C) = AB + AC c. (B+C) A = BA + CA (associative law of multiplication) (left distributive law) (right distributive law)
Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are defined. a. A(BC) = (AB)C b. A(B+C) = AB + AC c. (B+C) A = BA + CA (associative law of multiplication) (left distributive law) (right distributive law)
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
29
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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 uv and uv will appear later in the text.
27. Let u=
-2
3
-4
and v =
a
b
C
Compute uv, v u, uv, and
vu¹.
28. If u and v are in R", how are u' v and v' u related? How are
uvT 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
33.
34.
a¡1(b₁; + 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¡₁)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
the (column) definition of AIn.]
Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
Give a formula for (ABX), 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
each entry of the matrix).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cbdc46a-2c95-4576-b03b-d5c9a7d5376a%2F12a170fc-5e50-4481-a04d-37635a0ceb0d%2F7o79wn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:t
d
r
?
O
?
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 uv and uv will appear later in the text.
27. Let u=
-2
3
-4
and v =
a
b
C
Compute uv, v u, uv, and
vu¹.
28. If u and v are in R", how are u' v and v' u related? How are
uvT 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
33.
34.
a¡1(b₁; + 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¡₁)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
the (column) definition of AIn.]
Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
Give a formula for (ABX), 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
each entry of the matrix).
![t
d
r
?
O
?
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 uv and uv will appear later in the text.
27. Let u=
-2
3
-4
and v =
a
b
C
Compute uv, v u, uv, and
vu¹.
28. If u and v are in R", how are u' v and v' u related? How are
uvT 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
33.
34.
a¡1(b₁; + 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¡₁)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
the (column) definition of AIn.]
Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
Give a formula for (ABX), 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
each entry of the matrix).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cbdc46a-2c95-4576-b03b-d5c9a7d5376a%2F12a170fc-5e50-4481-a04d-37635a0ceb0d%2Frm9qis_processed.jpeg&w=3840&q=75)
Transcribed Image Text:t
d
r
?
O
?
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 uv and uv will appear later in the text.
27. Let u=
-2
3
-4
and v =
a
b
C
Compute uv, v u, uv, and
vu¹.
28. If u and v are in R", how are u' v and v' u related? How are
uvT 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
33.
34.
a¡1(b₁; + 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¡₁)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
the (column) definition of AIn.]
Prove Theorem 3(d). [Hint: Consider the jth row of (AB)T.]
Give a formula for (ABX), 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
each entry of the matrix).
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