13. An mxn upper triangular matrix is one whose entries below the main diagonal are 0's (as in Exercise 8). When is a square upper triangular matrix invertible? Justify your answer.
13. An mxn upper triangular matrix is one whose entries below the main diagonal are 0's (as in Exercise 8). When is a square upper triangular matrix invertible? Justify your answer.
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
13
![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.
om
7.
9. [M]
0
-3 -7
8
5
PAR
5
0
-3]
0
1 0
noutboo
-4
nodulo
-1 dm-3o 0
3 noin 502 08
-2in-63
-1 2
4
-6
7
5
6
10. [M] 7
9
2018
-1
set bail, 2.
3lo-57
2
7
-1
is sou
-9
0
0
-5
2
4.
Bord olib
of nois
1
-3
2
1
0 -7
1 11
10
3
6.
[9]
-7 0
1
0
8.
M:
0
0
1
0
-3
3 0 -1
2
0
3
5
0
0
-7
9
19 Dibos
-1
-5
3
missh
22 A
6
4
히
9
-4
4
0
7
9 6
6
4
47
28
0
10
8 T
bos
31 ob
7b (97
97d-x 29 e 2
4 2 81-8
1-8gb to toimun adı
3 10 9 9 wod noqon bas
6 490 1-5 to suelq ni bonu
5 2 11
5
4(M)
les
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.
11 a. If the equation Ax = 0 has only the trivial solution, then
A is row equivalent to the nxn identity matrix.
b. If the columns of A span R", then the columns are linearly
independent.
c. If A is an n x n matrix, then the equation Ax = b has at
least one solution for each b in R".
d. If the equation Ax = 0 has a nontrivial solution, then A
has fewer than n pivot positions.
e. If AT is not invertible, then A is not invertible.
12. A. If there is an n x n matrix D such that AD = I, then there
is also an n x n matrix C such that CA = I.
b. If the columns of A are linearly independent, then the
columns of A span R".
c. If the equation Ax = b has at least one solution for each
b in R", then the solution is unique for each b.
2.3 Characterizations of Invertible Matrices 117
d. If the linear transformation (x) → Ax maps R" into R",
then A has n pivot positions.
e. If there is a b in R" such that the equation Ax = b is
inconsistent, then the transformation x Ax is not one-
noitaup
to-one.
13. An mxn upper triangular matrix is one whose entries
below the main diagonal are 0's (as in Exercise 8). When
is a square upper triangular matrix invertible? Justify your
answer.
14. An mxn lower triangular matrix is one whose entries
w
above the main diagonal are 0's (as in Exercise 3). When
is a square lower triangular matrix invertible? Justify your
answer.
15. Can a square matrix with two identical columns be invert-
ible? Why or why not?
16. Is it possible for a 5 x 5 matrix to be invertible when its
columns do not span R5? Why or why not?
(S) bon (1)
17. If A is invertible, then the columns of A are linearly
independent. Explain why.molde sila od
18. If C is 6 x 6 and the equation Cx = v is consistent for every
v in R6, is it possible that for some v, the equation Cx= v
has more than one solution? Why or why not?
19. If the columns of a 7 x 7 matrix D are linearly independent,
what can you say about solutions of Dx = b? Why?t
20. If nxn matrices E and F have the property that EF = 1,
then E and F commute. Explain why.
28.
8089 30ITO.
21. If the equation Gx=y has more than one solution for some
y in R", can the columns of G span R"? Why or why not?
22. If the equation Hx = c is inconsistent for some c in R", what
can you say about the equation Hx=0? Why?
23. If an n x n matrix K cannot be row reduced to I, what can
you say about the columns of K? Why?
24. If L is n x n and the equation Lx = 0 has the trivial solution,
do the columns of L span R"? Why?
25. Verify the boxed statement preceding Example 1.
26. Explain why the columns of A² span R" whenever the
columns of A are linearly independent.
27. Show that if AB is invertible, so is A. You cannot use Theorem
6(b), because you cannot assume that A and B are invertible.
[Hint: There is a matrix W such that ABW= I. Why?]
Show that if AB is invertible, so is B.
29. If A is an n x n matrix and the equation Ax=b has more than
one solution for some b, then the transformation x→ Ax is
not one-to-one. What else can you say about this transforma-
tion? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcee71cba-b5b8-44a0-9e00-7f1c17b6ad3a%2F7eed0624-32ce-464f-9528-602d11339b6d%2F0fhzqhx_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
om
7.
9. [M]
0
-3 -7
8
5
PAR
5
0
-3]
0
1 0
noutboo
-4
nodulo
-1 dm-3o 0
3 noin 502 08
-2in-63
-1 2
4
-6
7
5
6
10. [M] 7
9
2018
-1
set bail, 2.
3lo-57
2
7
-1
is sou
-9
0
0
-5
2
4.
Bord olib
of nois
1
-3
2
1
0 -7
1 11
10
3
6.
[9]
-7 0
1
0
8.
M:
0
0
1
0
-3
3 0 -1
2
0
3
5
0
0
-7
9
19 Dibos
-1
-5
3
missh
22 A
6
4
히
9
-4
4
0
7
9 6
6
4
47
28
0
10
8 T
bos
31 ob
7b (97
97d-x 29 e 2
4 2 81-8
1-8gb to toimun adı
3 10 9 9 wod noqon bas
6 490 1-5 to suelq ni bonu
5 2 11
5
4(M)
les
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.
11 a. If the equation Ax = 0 has only the trivial solution, then
A is row equivalent to the nxn identity matrix.
b. If the columns of A span R", then the columns are linearly
independent.
c. If A is an n x n matrix, then the equation Ax = b has at
least one solution for each b in R".
d. If the equation Ax = 0 has a nontrivial solution, then A
has fewer than n pivot positions.
e. If AT is not invertible, then A is not invertible.
12. A. If there is an n x n matrix D such that AD = I, then there
is also an n x n matrix C such that CA = I.
b. If the columns of A are linearly independent, then the
columns of A span R".
c. If the equation Ax = b has at least one solution for each
b in R", then the solution is unique for each b.
2.3 Characterizations of Invertible Matrices 117
d. If the linear transformation (x) → Ax maps R" into R",
then A has n pivot positions.
e. If there is a b in R" such that the equation Ax = b is
inconsistent, then the transformation x Ax is not one-
noitaup
to-one.
13. An mxn upper triangular matrix is one whose entries
below the main diagonal are 0's (as in Exercise 8). When
is a square upper triangular matrix invertible? Justify your
answer.
14. An mxn lower triangular matrix is one whose entries
w
above the main diagonal are 0's (as in Exercise 3). When
is a square lower triangular matrix invertible? Justify your
answer.
15. Can a square matrix with two identical columns be invert-
ible? Why or why not?
16. Is it possible for a 5 x 5 matrix to be invertible when its
columns do not span R5? Why or why not?
(S) bon (1)
17. If A is invertible, then the columns of A are linearly
independent. Explain why.molde sila od
18. If C is 6 x 6 and the equation Cx = v is consistent for every
v in R6, is it possible that for some v, the equation Cx= v
has more than one solution? Why or why not?
19. If the columns of a 7 x 7 matrix D are linearly independent,
what can you say about solutions of Dx = b? Why?t
20. If nxn matrices E and F have the property that EF = 1,
then E and F commute. Explain why.
28.
8089 30ITO.
21. If the equation Gx=y has more than one solution for some
y in R", can the columns of G span R"? Why or why not?
22. If the equation Hx = c is inconsistent for some c in R", what
can you say about the equation Hx=0? Why?
23. If an n x n matrix K cannot be row reduced to I, what can
you say about the columns of K? Why?
24. If L is n x n and the equation Lx = 0 has the trivial solution,
do the columns of L span R"? Why?
25. Verify the boxed statement preceding Example 1.
26. Explain why the columns of A² span R" whenever the
columns of A are linearly independent.
27. Show that if AB is invertible, so is A. You cannot use Theorem
6(b), because you cannot assume that A and B are invertible.
[Hint: There is a matrix W such that ABW= I. Why?]
Show that if AB is invertible, so is B.
29. If A is an n x n matrix and the equation Ax=b has more than
one solution for some b, then the transformation x→ Ax is
not one-to-one. What else can you say about this transforma-
tion? Justify your answer.
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