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In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions.

In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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0 0 e fy its ar- Ax = b, and define v = w- p. Show that v, is a solution of Ax = 0. This shows that every solution of Ax = b has the form w=p+Vh, with p a particular solution of Ax = b and Vh a solution of Ax = 0. 26. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 27. Suppose A is the 3 x 3 zero matrix (with all zero entries). Describe the solution set of the equation Ax = 0. 28. If b 0, can the solution set of Ax = b be a plane through the origin? Explain.d In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions. 30. A is a 3 x 3 matrix with two pivot positions. 31. A is a 3 x 2 matrix with two pivot positions. 32. A is a 2 x 4 matrix with two pivot positions. -6 33. Given A= os noi -2 7 -3 quam 15 COPE 21 find one nontrivial solution of -9 15tted bas 1 2 -1 35. Cons 1 36. Cons 1 -2 1 moth 14 4 -5 -1 8 37. Cons 38. Supp 2MB the e exist uniqu with Ax = 0 by inspection. [Hint: Think of the equation Ax = 0 B A(u written as a vector equation.] A(cu 15geris tui mi equa origi o of A: Ax = 39. Let A fies t tor cu 40. Let SOLUTIONS TO PRACTICE PRO 1. Row reduce the augmented matri srl bol 0 8]-[ 9
Transcribed Image Text:0 0 e fy its ar- Ax = b, and define v = w- p. Show that v, is a solution of Ax = 0. This shows that every solution of Ax = b has the form w=p+Vh, with p a particular solution of Ax = b and Vh a solution of Ax = 0. 26. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 27. Suppose A is the 3 x 3 zero matrix (with all zero entries). Describe the solution set of the equation Ax = 0. 28. If b 0, can the solution set of Ax = b be a plane through the origin? Explain.d In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions. 30. A is a 3 x 3 matrix with two pivot positions. 31. A is a 3 x 2 matrix with two pivot positions. 32. A is a 2 x 4 matrix with two pivot positions. -6 33. Given A= os noi -2 7 -3 quam 15 COPE 21 find one nontrivial solution of -9 15tted bas 1 2 -1 35. Cons 1 36. Cons 1 -2 1 moth 14 4 -5 -1 8 37. Cons 38. Supp 2MB the e exist uniqu with Ax = 0 by inspection. [Hint: Think of the equation Ax = 0 B A(u written as a vector equation.] A(cu 15geris tui mi equa origi o of A: Ax = 39. Let A fies t tor cu 40. Let SOLUTIONS TO PRACTICE PRO 1. Row reduce the augmented matri srl bol 0 8]-[ 9
0 0 e fy its ar- Ax = b, and define v = w- p. Show that v, is a solution of Ax = 0. This shows that every solution of Ax = b has the form w=p+Vh, with p a particular solution of Ax = b and Vh a solution of Ax = 0. 26. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 27. Suppose A is the 3 x 3 zero matrix (with all zero entries). Describe the solution set of the equation Ax = 0. 28. If b 0, can the solution set of Ax = b be a plane through the origin? Explain.d In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions. 30. A is a 3 x 3 matrix with two pivot positions. 31. A is a 3 x 2 matrix with two pivot positions. 32. A is a 2 x 4 matrix with two pivot positions. -6 33. Given A= os noi -2 7 -3 quam 15 COPE 21 find one nontrivial solution of -9 15tted bas 1 2 -1 35. Cons 1 36. Cons 1 -2 1 moth 14 4 -5 -1 8 37. Cons 38. Supp 2MB the e exist uniqu with Ax = 0 by inspection. [Hint: Think of the equation Ax = 0 B A(u written as a vector equation.] A(cu 15geris tui mi equa origi o of A: Ax = 39. Let A fies t tor cu 40. Let SOLUTIONS TO PRACTICE PRO 1. Row reduce the augmented matri srl bol 0 8]-[ 9
Transcribed Image Text:0 0 e fy its ar- Ax = b, and define v = w- p. Show that v, is a solution of Ax = 0. This shows that every solution of Ax = b has the form w=p+Vh, with p a particular solution of Ax = b and Vh a solution of Ax = 0. 26. Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 27. Suppose A is the 3 x 3 zero matrix (with all zero entries). Describe the solution set of the equation Ax = 0. 28. If b 0, can the solution set of Ax = b be a plane through the origin? Explain.d In Exercises 29-32, (a) does the equation Ax = 0 have a nontriv- ial solution and (b) does the equation Ax = b have at least one solution for every possible b? 29. A is a 3 x 3 matrix with three pivot positions. 30. A is a 3 x 3 matrix with two pivot positions. 31. A is a 3 x 2 matrix with two pivot positions. 32. A is a 2 x 4 matrix with two pivot positions. -6 33. Given A= os noi -2 7 -3 quam 15 COPE 21 find one nontrivial solution of -9 15tted bas 1 2 -1 35. Cons 1 36. Cons 1 -2 1 moth 14 4 -5 -1 8 37. Cons 38. Supp 2MB the e exist uniqu with Ax = 0 by inspection. [Hint: Think of the equation Ax = 0 B A(u written as a vector equation.] A(cu 15geris tui mi equa origi o of A: Ax = 39. Let A fies t tor cu 40. Let SOLUTIONS TO PRACTICE PRO 1. Row reduce the augmented matri srl bol 0 8]-[ 9
Expert Solution
Step 1

Introduction:

Pivots in a matrix;

  1. Any row that does not contain only zeros its first non zero number is a 1 also is called the leading 1. These are also called pivots.
  2. For two successive rows with leading 1's, the 1 in the lower row is to the right of the 1 in the upper row. 

Given:  A is a 3×3 matrix with three pivot positions.

To find:

Justify the solutions for the system.

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