Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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17
![2.3 EXERCISES
Unless otherwise specified, assume that all matrices in these
exercises are n xn. Determine which of the matrices in Exercises
1-10 are invertible. Use as few calculations as possible. Justify
your answers.
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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 n x n identity matrix.
b. If the columns of A span R", then the columns are linearly
independent.
AP
c. If A is an n x n matrix, then the equation Ax = b has at
least one solution for each b in R". Yoon
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.
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".
Jorn
.
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-
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
BMoldinavill
answer.
14. An mxn lower triangular matrix is one whose entries
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?
2500
16. Is it possible for a 5 x 5 matrix to be invertible when its
columns do not span R5? Why or why not?
17. If A is invertible, then the columns of A are linearly
independent. Explain why.
HU
ODAC 201
06
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?
020qqu2 TE
19. If the columns of a 7 x 7 matrix D are linearly independent,
what can you say about solutions of Dx = b? Why?lot
20. If n ×n matrices E and F have the property that EF = I,
then E and F commute. Explain why.
gai
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 In, what can
you say about the columns of K? Why?
If Lis n x n and the equation Lx = 0 has the trivial solution,
do the columns of L span R"? Why? bo
24.
bo
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?]
28.
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.](https://content.bartleby.com/qna-images/question/046862d2-b2b1-4a60-bae0-248c79b95a0e/74c74f7d-7bda-49d4-9bc9-7158b675f606/ged2li_processed.jpeg)
Transcribed Image Text:2.3 EXERCISES
Unless otherwise specified, assume that all matrices in these
exercises are n xn. Determine which of the matrices in Exercises
1-10 are invertible. Use as few calculations as possible. Justify
your answers.
ad
5
1.
3.
5.
7.
5
-3
co
3
-2
7
-6
9. [M]
cod
0
-7
0
(5 9-1
0
3
stw n
alo-5th tot
ain!
1 0
2
6.
purblies oli song of mo
-4 -9
7
10. [M]
0
-1-300 120 of
5 8 ugu-3
-6301 2.
001-1pm 2
qs 1
−1
4 b 0 -7
-6 1 1 11
adi
7 -5
200 b2
-1
-4 6
ONT
6-9
[ ]
5
3
1
6
4
7
21
5 3
9 64 4
8 5 2
2.
4.
Tollib
10 d
3
در
8.
1-71
9
19
-1
-7
3
2
[!
0
7
9
8 -8
10 9
-9 -5
11 4
1011
-3
1
0
000
0
-5
3
6
1 3 7
002
4
−1
9
-4
0
4
596
8
0 0 10
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 n x n identity matrix.
b. If the columns of A span R", then the columns are linearly
independent.
AP
c. If A is an n x n matrix, then the equation Ax = b has at
least one solution for each b in R". Yoon
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.
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".
Jorn
.
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-
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
BMoldinavill
answer.
14. An mxn lower triangular matrix is one whose entries
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?
2500
16. Is it possible for a 5 x 5 matrix to be invertible when its
columns do not span R5? Why or why not?
17. If A is invertible, then the columns of A are linearly
independent. Explain why.
HU
ODAC 201
06
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?
020qqu2 TE
19. If the columns of a 7 x 7 matrix D are linearly independent,
what can you say about solutions of Dx = b? Why?lot
20. If n ×n matrices E and F have the property that EF = I,
then E and F commute. Explain why.
gai
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 In, what can
you say about the columns of K? Why?
If Lis n x n and the equation Lx = 0 has the trivial solution,
do the columns of L span R"? Why? bo
24.
bo
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?]
28.
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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