Unless otherwise specified, assume that all matrices in these exercises are nxn. Determine which of the matrices in Exercises 1-10 are invertible. Use as few calculations as possible. Justify your answers. 1. 3. 5. 5 -3 5 -3 8 7 -6 0 -7 5 0 0 03 1 0 -4 -9 -1 3-5 2 7 sou 2. 4. 201 6. -4 6 6 -7 0 3 0 20 -1 9 1 -5 -4 1611 1] 0 3 4 -3 0
Unless otherwise specified, assume that all matrices in these exercises are nxn. Determine which of the matrices in Exercises 1-10 are invertible. Use as few calculations as possible. Justify your answers. 1. 3. 5. 5 -3 5 -3 8 7 -6 0 -7 5 0 0 03 1 0 -4 -9 -1 3-5 2 7 sou 2. 4. 201 6. -4 6 6 -7 0 3 0 20 -1 9 1 -5 -4 1611 1] 0 3 4 -3 0
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 10AEXP
Related questions
Question
5
![cab
Guide
em 5.
35.
-1
j = 1,..., n, let a,, b,, and e, denote the j th columns of
A, B, and I, respectively. Use the facts that a; -aj+1 =
and bj = ej-ej+1 for j = 1,...,n - 1, and
ej
an = bn = en.
3
-6
4
· [₁
37. C =
Find this by row reducing [A
13]
0
39. 27,.30, and .23 inch, respectively
41. [M] 12, 1.5, 21.5, and 12 newtons, respectively
1
0
0
Section 2.3, page 117
The abbreviation IMT (here and in the Study Guide) denotes the
Invertible Matrix Theorem (Theorem 8)
e3 ].
1. Invertible, by the IMT. Neither column of the matrix is a
multiple of the other column, so they are linearly
independent. Also, the matrix is invertible by Theorem 4 in
Section 2.2 because the determinant is nonzero.
3. Invertible, by the IMT. The matrix row reduces to
5
0
0
0
-7
0 and has 3 pivot positions.
0
3
0
5. Not invertible, by the IMT. The matrix row reduces to
2
-5
0
and is not row equivalent to 13.
19. By
Dx
Car
21. The
2.2
IM
spa
23. Sta
an
de
25. H
27. L
A
29. S
S
t
31.
no
b
33.
35.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcee71cba-b5b8-44a0-9e00-7f1c17b6ad3a%2Ff1623f84-d74f-4338-ad9d-e5a764626510%2Fiz5i8mn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:cab
Guide
em 5.
35.
-1
j = 1,..., n, let a,, b,, and e, denote the j th columns of
A, B, and I, respectively. Use the facts that a; -aj+1 =
and bj = ej-ej+1 for j = 1,...,n - 1, and
ej
an = bn = en.
3
-6
4
· [₁
37. C =
Find this by row reducing [A
13]
0
39. 27,.30, and .23 inch, respectively
41. [M] 12, 1.5, 21.5, and 12 newtons, respectively
1
0
0
Section 2.3, page 117
The abbreviation IMT (here and in the Study Guide) denotes the
Invertible Matrix Theorem (Theorem 8)
e3 ].
1. Invertible, by the IMT. Neither column of the matrix is a
multiple of the other column, so they are linearly
independent. Also, the matrix is invertible by Theorem 4 in
Section 2.2 because the determinant is nonzero.
3. Invertible, by the IMT. The matrix row reduces to
5
0
0
0
-7
0 and has 3 pivot positions.
0
3
0
5. Not invertible, by the IMT. The matrix row reduces to
2
-5
0
and is not row equivalent to 13.
19. By
Dx
Car
21. The
2.2
IM
spa
23. Sta
an
de
25. H
27. L
A
29. S
S
t
31.
no
b
33.
35.
![Or
e
is in
A is
inear
from
nt (by
, and
ill-
of
Lice
ТОГ
are
lar.
2.3 EXERCISES
Unless otherwise specified, assume that all matrices in these
exercises are nxn. Determine which of the matrices in Exercises
1-10 are invertible. Use as few calculations as possible. Justify
your answers.
1.
3.
5.
7.
5
7
-3 -6
5
0
-3 -7
8
010 3-5
1
00
-4
-1-3
35
lub d'
9. [M]
nostab
to bel
-9
10. [M]
0
5 -1
4
-6
-2 -6 3 2
0 -1
-1
5 3
6
7
9
7 -5
8832
4565
000
8 5
2
7
8
152
17
-3
1
2
NE
0 -7
11
10
3
1
ci
4.
for batunado
6.
의
9
19
-1
7
8
100
6 -9
-7 0
3 0
20
4 -9
11
2
160
1
0
-3
1
0
9
-8
9
-5
4
0
0
536
-5
3
5
0
0 0
7920
4
-1
9
440
-4
4
680
8
sil
2.3 Charact
In Exercises 11 and 12, the matrices are all n xn. Each part of
the exercises is an implication of the form "If "statement 1",
then "statement 2"." Mark an implication as True if the truth of
"statement 2" always follows whenever "statement 1" happens
to be true. An implication is False if there is an instance in
which "statement 2" is false but "statement 1" is true. Justify each
answer.
d. If the linear t
then A has n
13. An mxn upp
below the main
is a square upp
answer.
Intion 4x = 0 has only the trivial solution, then
trix
B
e. If there is a
inconsistent,
to-one.
14. An mxn lov
above the ma
is a square lo
answer.
15. Can a square
ible? Why
16. Is it possible
columns do
17. If A is inve
independent
461
18. If C is 6 x6
v in R6, is
has more th
19. If the colu
what can y
20. If nxnn
then E anc
21. If the equ
y in R"..
22. If the equ
can you
23. If an nx
you say
If I is n
24](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcee71cba-b5b8-44a0-9e00-7f1c17b6ad3a%2Ff1623f84-d74f-4338-ad9d-e5a764626510%2Fqpi803s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Or
e
is in
A is
inear
from
nt (by
, and
ill-
of
Lice
ТОГ
are
lar.
2.3 EXERCISES
Unless otherwise specified, assume that all matrices in these
exercises are nxn. Determine which of the matrices in Exercises
1-10 are invertible. Use as few calculations as possible. Justify
your answers.
1.
3.
5.
7.
5
7
-3 -6
5
0
-3 -7
8
010 3-5
1
00
-4
-1-3
35
lub d'
9. [M]
nostab
to bel
-9
10. [M]
0
5 -1
4
-6
-2 -6 3 2
0 -1
-1
5 3
6
7
9
7 -5
8832
4565
000
8 5
2
7
8
152
17
-3
1
2
NE
0 -7
11
10
3
1
ci
4.
for batunado
6.
의
9
19
-1
7
8
100
6 -9
-7 0
3 0
20
4 -9
11
2
160
1
0
-3
1
0
9
-8
9
-5
4
0
0
536
-5
3
5
0
0 0
7920
4
-1
9
440
-4
4
680
8
sil
2.3 Charact
In Exercises 11 and 12, the matrices are all n xn. Each part of
the exercises is an implication of the form "If "statement 1",
then "statement 2"." Mark an implication as True if the truth of
"statement 2" always follows whenever "statement 1" happens
to be true. An implication is False if there is an instance in
which "statement 2" is false but "statement 1" is true. Justify each
answer.
d. If the linear t
then A has n
13. An mxn upp
below the main
is a square upp
answer.
Intion 4x = 0 has only the trivial solution, then
trix
B
e. If there is a
inconsistent,
to-one.
14. An mxn lov
above the ma
is a square lo
answer.
15. Can a square
ible? Why
16. Is it possible
columns do
17. If A is inve
independent
461
18. If C is 6 x6
v in R6, is
has more th
19. If the colu
what can y
20. If nxnn
then E anc
21. If the equ
y in R"..
22. If the equ
can you
23. If an nx
you say
If I is n
24
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