ve Theorem 4 for A 25. Show that if ad bc = 0, then the equation Ax = 0 has more than one solution. Why does this imply that A is not invertible? [Hint: First, consider a = b = 0. Then, if a and - 36. [L th
ve Theorem 4 for A 25. Show that if ad bc = 0, then the equation Ax = 0 has more than one solution. Why does this imply that A is not invertible? [Hint: First, consider a = b = 0. Then, if a and - 36. [L th
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
25
![17. Solve the equation AB = BC for A, assuming
Care square and B is invertible.
18. Suppose P is invertible and A = PBP-1. Solve for B in
terms of A.
19. If A, B, and C are n x n invertible matrices, does the equation
C-¹(A + X)B¯¹ = 1, have a solution, X? If so, find it.
20. Suppose A, B, and X are nxn matrices with A, X, and
A - AX invertible, and suppose
(A - AX)¹ = X-¹ B
a. Explain why B is invertible.
b. Solve (3) for X. If you need to invert a matrix, explain
why that matrix is invertible.
(3)
22. Explain why the columns of an n x n matrix A span R" when
A is invertible. [Hint: Review Theorem 4 in Section 1.4.]
21. Explain why the columns of an n x n matrix A are linearly 34. Repeal
independent when A is invertible.
23. Suppose A is n x n and the equation Ax = 0 has only the
trivial solution. Explain why A has n pivot columns and A is
row equivalent to I. By Theorem 7, this shows that A must
be invertible. (This exercise and Exercise 24 will be cited in
Section 2.3.)
24. Suppose A is n x n and the equation Ax = b has a solution
for each b in R". Explain why A must be invertible. [Hint: Is
A row equivalent to I,?]
Exercises 25 and 26 prove Theorem 4 for A =
а
-[²2]
C
d
25. Show that if ad - bc = 0, then the equation Ax = 0 has
more than one solution. Why does this imply that A is not
invertible? [Hint: First, consider a = b = 0. Then, if a and
-b
b are not both zero, consider the vector x =
a
26. Show that if ad - bc #0, the formula for A-1
works.
1
31. X-23
33. Use the
Exercises 27 and 28 prove special cases of the facts about elemen-
tary matrices stated in the box following Example 5. Here A is a
Let A
inverse
and Br
A =
correc
35. Let A
38.
witho
36. [M] L
third
37. Let
error
Com](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F286f3f44-ea77-4623-ae9f-381dd0d9eeb1%2F0de7b4d0-5c41-48d2-9ae9-59142efe4be0%2F54utmf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:17. Solve the equation AB = BC for A, assuming
Care square and B is invertible.
18. Suppose P is invertible and A = PBP-1. Solve for B in
terms of A.
19. If A, B, and C are n x n invertible matrices, does the equation
C-¹(A + X)B¯¹ = 1, have a solution, X? If so, find it.
20. Suppose A, B, and X are nxn matrices with A, X, and
A - AX invertible, and suppose
(A - AX)¹ = X-¹ B
a. Explain why B is invertible.
b. Solve (3) for X. If you need to invert a matrix, explain
why that matrix is invertible.
(3)
22. Explain why the columns of an n x n matrix A span R" when
A is invertible. [Hint: Review Theorem 4 in Section 1.4.]
21. Explain why the columns of an n x n matrix A are linearly 34. Repeal
independent when A is invertible.
23. Suppose A is n x n and the equation Ax = 0 has only the
trivial solution. Explain why A has n pivot columns and A is
row equivalent to I. By Theorem 7, this shows that A must
be invertible. (This exercise and Exercise 24 will be cited in
Section 2.3.)
24. Suppose A is n x n and the equation Ax = b has a solution
for each b in R". Explain why A must be invertible. [Hint: Is
A row equivalent to I,?]
Exercises 25 and 26 prove Theorem 4 for A =
а
-[²2]
C
d
25. Show that if ad - bc = 0, then the equation Ax = 0 has
more than one solution. Why does this imply that A is not
invertible? [Hint: First, consider a = b = 0. Then, if a and
-b
b are not both zero, consider the vector x =
a
26. Show that if ad - bc #0, the formula for A-1
works.
1
31. X-23
33. Use the
Exercises 27 and 28 prove special cases of the facts about elemen-
tary matrices stated in the box following Example 5. Here A is a
Let A
inverse
and Br
A =
correc
35. Let A
38.
witho
36. [M] L
third
37. Let
error
Com
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