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In Exercises 11 and 12, the matrices are all nxn. 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 I 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. 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.

In Exercises 11 and 12, the matrices are all nxn. 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 I 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. 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.

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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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 O's (as in Exercise 8). When is a square upper triangular matrix invertible? Justify your answer. I 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?
Transcribed Image Text: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 O's (as in Exercise 8). When is a square upper triangular matrix invertible? Justify your answer. I 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?
In Exercises 11 and 12, the matrices are all nxn. 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 I 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. 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 A¹ 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. 20. If n x n matrices E and then E and F commute. 21. If the equation Gx = y y in R", can the column 22. If the equation Hx = ci can you say about the ec If an n x n matrix K can you say about the colum If L is n x n and the equa do the columns of L span 23. 24. 25. Verify the boxed stateme 26. Explain why the colum columns of A are linearly 27. Show that if AB is invertib 6(b), because you cannot [Hint: There is a matrix F 28. Show that if AB is inverti 29. If A is an n x n matrix and one solution for some b, not one-to-one. What else tion? Justify your answer.
Transcribed Image Text:In Exercises 11 and 12, the matrices are all nxn. 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 I 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. 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 A¹ 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. 20. If n x n matrices E and then E and F commute. 21. If the equation Gx = y y in R", can the column 22. If the equation Hx = ci can you say about the ec If an n x n matrix K can you say about the colum If L is n x n and the equa do the columns of L span 23. 24. 25. Verify the boxed stateme 26. Explain why the colum columns of A are linearly 27. Show that if AB is invertib 6(b), because you cannot [Hint: There is a matrix F 28. Show that if AB is inverti 29. If A is an n x n matrix and one solution for some b, not one-to-one. What else tion? Justify your answer.
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