not and en- -0 37. Let A = 2 3 5 Construct a 2 x 3 matrix C (by trial and error) using only 1,-1, and 0 as entries, such that CA = 1₂. Compute AC and note that AC # 13.
not and en- -0 37. Let A = 2 3 5 Construct a 2 x 3 matrix C (by trial and error) using only 1,-1, and 0 as entries, such that CA = 1₂. Compute AC and note that AC # 13.
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
37
![ix Algebra
~1/ X], then X = A¹B.
.. then row reduction of [A B] is much
both A¹ and AB.
here B and Caren xp matrices and A
B = C. Is this true, in general, when
0, where B and Care in x # matrices
w that B = C.
invertible x n matrices. Show that
by producing a matrix D such that
ABC) = 1.
xn, B is invertible, and AB is invert-
rtible. [Hinr: Let C = AB, and solve
= BC for A, assuming that A, B, and
vertible.
and A = PBP. Solve for B in
nvertible matrices, does the equation
have a solution, X? If so, find it.
rex matrices with A, X, and
suppose
ertible.
u need to invert a matrix, explain
ertible.
of an n x n matrix A are linearly
vertible.
(3)
anx matrix A span R" when
iew Theorem 4 in Section 1.4.]
he equation Ax = 0 has only the
by A has n pivot columns and A is
heorem 7, this shows that A must
e and Exercise 24 will be cited in
e equation Ax=b has a solution
hy A must be invertible. [Hint: Is
= [²2]
C
d
rem 4 for A =
then the equation Ax = 0 has
ny does this imply that A is not
sider a = b = 0. Then, if á and
= [-]₁
a
ne formula for A works.
the vector X =
cases of the facts about elemen-
lowing Example 5. Here A is a
3 x 3 matrix and I = 13. (A general proof would require slightly
more notation.)
27. a. Use equation (1)
row, (A) = row, (I)
29.
b. Show that if rows 1 and 2 of A are interchanged, then the
result may be written as EA, where E is an elementary
matrix formed by interchanging rows 1 and 2 of 1.
28. Show that if row 3 of A is replaced by row3 (A) - 4-row₁(4).
the result is EA, where E is formed from / by replacing
row3 (7) by row3 (1)-4 row, (I).
31.
c. Show that if row 3 of A is multiplied by 5, then the result
may be written as EA, where E is formed by multiplying
row 3 of 1 by 5.
Find the inverses of the matrices in Exercises 29-32, if they exist.
Use the algorithm introduced in this section.
[43]
1
-3
1
1
1
0-2
1 4
2 -3 4
A =
0
1
1 I
correct.
0]
32.
32.122
-2
33. Use the algorithm from this section to find the inverses of
0 0 0
0
0
1
35. Let A =
from Section 2.1 to show that
A, for i = 1.2, 3.
0
2
2
♦
and
37. Let A =
2 3
36. [M] Let A =
0
0
3
a
third columns of A
2
3
1 5
30.
Let A be the corresponding n x n matrix, and let B be its
inverse. Guess the form of B, and then prove that AB = I
and BA = 1.
1
34. Repeat the strategy of Exercise 33 to guess the inverse of
1
01
1
I
..
1
1
['$
4
20
1
1
5 10
¹9]
7
0
0
11
1 -2
T
3
6 -4
-2 -7 -9
2 5 6 . Find the third column of A
1 3 4
without computing the other columns.
0
9]
4 -7
0
1
I
-25 -9 -27
546 180
537 . Find the second and
154 50 149
without computing the first column.
Construct a 2 x 3 matrix C (by trial and
error) using only 1,-1, and 0 as entries, such that CA = 1₂.
Compute AC and note that AC # 13.
38. Let A = [
II
1
Prove that your guess is
Construct a 4 x 2 matrix D
using only 1 and 0 as entries
ble that CA = 1₁ for some =
.005
002
,002 .004
.001 .002
with flexibility measured
that forces of 30, 50, ar
2, and 3. respectively, in
corresponding deflections
39. Let D =
40. [M] Compute the stiffnes
List the forces needed to
point 3, with zero deflec
41. [M] Let D =
.0040
.0030
0010
.0005
2.3 CHAR](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F286f3f44-ea77-4623-ae9f-381dd0d9eeb1%2F694cbe39-e115-40d7-bd37-10c56bf27a84%2F3znj3gb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ix Algebra
~1/ X], then X = A¹B.
.. then row reduction of [A B] is much
both A¹ and AB.
here B and Caren xp matrices and A
B = C. Is this true, in general, when
0, where B and Care in x # matrices
w that B = C.
invertible x n matrices. Show that
by producing a matrix D such that
ABC) = 1.
xn, B is invertible, and AB is invert-
rtible. [Hinr: Let C = AB, and solve
= BC for A, assuming that A, B, and
vertible.
and A = PBP. Solve for B in
nvertible matrices, does the equation
have a solution, X? If so, find it.
rex matrices with A, X, and
suppose
ertible.
u need to invert a matrix, explain
ertible.
of an n x n matrix A are linearly
vertible.
(3)
anx matrix A span R" when
iew Theorem 4 in Section 1.4.]
he equation Ax = 0 has only the
by A has n pivot columns and A is
heorem 7, this shows that A must
e and Exercise 24 will be cited in
e equation Ax=b has a solution
hy A must be invertible. [Hint: Is
= [²2]
C
d
rem 4 for A =
then the equation Ax = 0 has
ny does this imply that A is not
sider a = b = 0. Then, if á and
= [-]₁
a
ne formula for A works.
the vector X =
cases of the facts about elemen-
lowing Example 5. Here A is a
3 x 3 matrix and I = 13. (A general proof would require slightly
more notation.)
27. a. Use equation (1)
row, (A) = row, (I)
29.
b. Show that if rows 1 and 2 of A are interchanged, then the
result may be written as EA, where E is an elementary
matrix formed by interchanging rows 1 and 2 of 1.
28. Show that if row 3 of A is replaced by row3 (A) - 4-row₁(4).
the result is EA, where E is formed from / by replacing
row3 (7) by row3 (1)-4 row, (I).
31.
c. Show that if row 3 of A is multiplied by 5, then the result
may be written as EA, where E is formed by multiplying
row 3 of 1 by 5.
Find the inverses of the matrices in Exercises 29-32, if they exist.
Use the algorithm introduced in this section.
[43]
1
-3
1
1
1
0-2
1 4
2 -3 4
A =
0
1
1 I
correct.
0]
32.
32.122
-2
33. Use the algorithm from this section to find the inverses of
0 0 0
0
0
1
35. Let A =
from Section 2.1 to show that
A, for i = 1.2, 3.
0
2
2
♦
and
37. Let A =
2 3
36. [M] Let A =
0
0
3
a
third columns of A
2
3
1 5
30.
Let A be the corresponding n x n matrix, and let B be its
inverse. Guess the form of B, and then prove that AB = I
and BA = 1.
1
34. Repeat the strategy of Exercise 33 to guess the inverse of
1
01
1
I
..
1
1
['$
4
20
1
1
5 10
¹9]
7
0
0
11
1 -2
T
3
6 -4
-2 -7 -9
2 5 6 . Find the third column of A
1 3 4
without computing the other columns.
0
9]
4 -7
0
1
I
-25 -9 -27
546 180
537 . Find the second and
154 50 149
without computing the first column.
Construct a 2 x 3 matrix C (by trial and
error) using only 1,-1, and 0 as entries, such that CA = 1₂.
Compute AC and note that AC # 13.
38. Let A = [
II
1
Prove that your guess is
Construct a 4 x 2 matrix D
using only 1 and 0 as entries
ble that CA = 1₁ for some =
.005
002
,002 .004
.001 .002
with flexibility measured
that forces of 30, 50, ar
2, and 3. respectively, in
corresponding deflections
39. Let D =
40. [M] Compute the stiffnes
List the forces needed to
point 3, with zero deflec
41. [M] Let D =
.0040
.0030
0010
.0005
2.3 CHAR
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