1. Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher

2. Friday, Feb 7 Chapter 2 No hand-in-homework assignment Main Idea: I do not want any surprises on the test. Key Words: Practice test Goal: Test over the material taught in class.

3. 1. The function T|x| = |x-y| is a |y| |y-x| linear transformation. True. It has matrix | 1 -1 |. |-1 1 |

4. 2. Matrix | 1/2 -1/2 | represents a | 1/2 1/2 | rotation. False (1/2) 2 + (1/2) 2 = 1/2 =/= 1

5. 3. If A is any invertible nxn matrix, then rref(A) = In. True. A matrix is invertible if and only if its RCF is the identity.

6. 4. The formula (A 2 ) -1 = (A -1 ) 2 holds for all invertible matrices A. True. A A A -1 A -1 = I.

7. 5. The formula AB=BA holds for all nxn matrices A and B. False. | 0 1| |0 0| =/= | 0 0 | | 0 1 | | 0 0| |1 0| | 1 0 | | 0 0 |

8. 6. If AB = In for two nxn matrices A and B, then A must be the inverse of B. True. This is false if A and B are not square.

9. 7. If A is a 3x4 matrix and B is a 4x5 matrix, then AB will be a 5x3 matrix. False. AB will be a 3x5 matrix.

10. 8. The function T|x| = |y| is a linear |y| |1| transformation. False. T (2 |0|) = |0| =/= 2 T|0| = | 0 | |0| |1| |0| | 2 |

11. 9. The matrix | 5 6 | represents a |-6 5 | rotation-dilation. True. The dilation is by Sqrt[61] the angle is ArcTan[-6/5] = -0.876058 radians

12. 10. If A is any invertible nxn matrix, then A commutes with A -1. True. By definition, A A -1 = A -1 A = I

13. 11. Matrix | 1 2 | is invertible. | 3 6 | False. The RCF is | 1 2 |. | 0 0 |

14. | 1 1 1 | Matrix | 1 0 1 | is invertible. | 1 1 0 | True. | 1 1 1 | | 1 0 1| | 1 0 0 | | 1 0 1 | ~ | 0 1 0| ~ | 0 1 0 | | 1 1 0 | | 0 1 -1| | 0 0 1 |

15. 13. There is an upper triangular 2x2 matrix A such that A 2 = | 1 1 | | 0 1 | True. A = | 1 1/2 | is one possibility. | 0 1 |

16. 14. The function T|x| = |(y+1) 2 – (y-1) 2 | is a linear |y| |(x-3) 2 – (x+3) 2 | transformation. True. T|x| = | 4 y|. |y| |-12 x|

17. 15. Matrix | k -2 | is invertible for all | 5 k-6 | real numbers k. True. | k -2 | ~ | 1 (k-6)/5 | ~ | 1 (k-6)/5 | | 5 k-6 | | k -2 | | 0 (-k^2+6k-10)/5| This polynomial has roots 3 (+/-) i so for all REAL numbers k, the RCF is I and it is invertible.

18. 16. There is a real number k such that the matrix | k-1 -2 | fails to be invertible. | -4 k-3 | True. k = -1 | -2 -2 | k = 5 | 4 -2 |. | -4 -4 | | -4 2 |

19. 17. There is a real number k such that the matrix | k-2 3 | fails to be | -3 k-2 | invertible. False. | k-2 3 | ~ | 1 -(k-2)/3 | ~ | 1 -(k-2)/3 | | -3 k-2 | | k-2 3 | | 0 (k-2) 2 +3| the roots are k = 2 (+/-) i Sqrt[3] which are not real.

20. 18. Matrix | -0.6 0.8 | represents a |-0.8 -0.6 | rotation. True: theta = Pi + ArcCos[0.6] = 4.06889

21. 19. The formula det(2A) = 2 det(A) holds for all 2x2 matrices A. False. det(2A) = 4 det(A).

22. 20. There is a matrix A such that | 1 2 | A | 5 6 | = | 1 1 |. | 3 4 | | 7 8 | | 1 1 | True | 1 2 | -1 | 1 1 | | 5 6 | |-1 | 3 4 | | 1 1 | | 7 8 | Should work. 1/2 | 1 -1 | | -1 1 |

23. 21. There is a matrix A such that A | 1 1 | = | 1 2 |. | 1 1 | | 1 2 | False Any linear combination of the rows of | 1 1 | will look like | x x |. | 1 1 | | y y |

24. 22. There is a matrix A such that | 1 2 | A = | 1 1 |, | 1 2 | | 1 1 | True. | 1 1 | works. | 0 0 |

25. 23. Matrix | -1 2 | represents a shear. | -2 3 | False | -1 2 | |x| = | -x + 2y| = |x| +2(-x+y) | 1| | -2 3 | |y| | -2x+3y| |y| | 1| The fixed vector has | 1 |. | 1 |

26. 24. | 1 k | 3 = | 1 3k | for all real | 0 1 | | 0 1 | numbers k. True:

27. 25. The matrix product | a b | | d -b | is always a scalar | c d | | -c a | of I 2. True. The scalar is ad-bc.

28. 26. There is a nonzero upper triangular 2x2 matrix A such that A 2 = | 0 0 |. | 0 0 | True. A = | 0 1 | is one possibility. | 0 0 |

29. 27. There is a positive integer n such that | 0 -1 | n = I 2. | 1 0 | True. n = 4 is one possibility.

30. 28. There is an invertible 2x2 matrix A such that A -1 = | 1 1 |. | 1 1 | False. The RCF of | 1 1 | = | 1 1 | | 1 1 | | 0 0 | so | 1 1 | cannot be an invertible matrix. | 1 1 |

31. 29. There is an invertible nxn matrix with two identical rows. False. If A has two identical rows, then AB has 2 identical rows also. Thus AB cannot be I.

32. 30. If A 2 = I n, then matrix A must be invertible. True. In fact, A is its own inverse.

33. 31. If A 17 = I 2, then A must be I 2. False A = | Cos[t] -Sin[t] | | Sin[t] Cos[t] | Where t = 2 Pi/17 should work.

34. 32. If A 2 = I 2, then A must be either I 2 or –I 2. False A = | -1 0 | is one possibility. | 0 1 |

35. 33. If matrix A is invertible, then matrix 5 A is invertible as well. True. And (5A) -1 = 1/5 A -1.

36. 34. If A and B are two 4x3 matrices such that AV = BV for all vectors v in R 3, then matrices A and B must be equal. True. It follows that AI = BI for the 3x3 identity matrix I. Thus A=B.

37. 35. If matrices A and B commute, then the formula A 2 B = BA 2 must hold. True. A 2 B = AAB = ABA=BAA=BA 2.

38. 36. If A 2 = A for an invertible nxn matrix A, then A must be I n. True. Multiply through by A -1 giving A=I.

39. 37. If matrices A and B are both invertible, then matrix A+B must be invertible as well. False. Let B = -A.

40. 38. The equation A 2 = A holds for all 2x2 matrices A representing an orthogonal projection. True. Once you have projected once by A, subequent actions by A will simply fix the vector.

41. 39. If matrix | a b c | is invertible, then | d e f | | g h I | matrix | a b | must be invertible as well. | d e | | 0 0 1 | False. | 0 1 0 | Is an example. | 1 0 0 |

42. 40. If A 2 is invertible, then matrix A itself must be invertible. True. For A 2 to be defined, then A must be square. If AAB = I, then A must be right invertible so A is invertible.

43. 41. The equation A -1 = A holds for all 2x2 matrices A representing a reflection. True. For a reflection A 2 = I.

44. 42. The formula (AV).(AW) = V.W holds for all invertible 2x2 matrices A and for all vectors V and W in R 2. False. | 1 1 | | 0 |.| 1 1 | | 1 | = 1 | 0 1 | | 1 | | 0 1| | 0 |

45. 43. There exist a 2x3 matrix A and a 3x2 matrix B such that AB = I 3. True. | 1 0 0 | | 1 0 | = | 1 0 | | 0 1 0 | | 0 1| | 0 1 | | 0 0|

46. 44. There exist a 3x2 matrix A and a 2x3 matrix B such that AB = I 3. False. There must be some X =/= 0 such that BX = 0. Then 0 = ABX = X. Contradiction.

47. 45. If A 2 + 3A + 4 I 3 = 0 for a 3x3 matrix A then A must be invertible. True. A(A+3) = -4 I 3 so the inverse of A is (-1/4)(A+3).

48. 46. If A is an nxn such that A 2 = 0, then matrix I n +A must be invertible. True. (I n +A)(I n -A) = I.

49. 47. If matrix A represents a shear, then the formula A 2 -2A+I 2 = 0 must hold. True. (A-I)X will be a fixed vector. So A(A-I)X = (A-I)X which means A2-2A+I = 0.

50. 48. If T is any linear transformation from R 3 to R 3, then T(VxW) = T(V)xT(W) for all vectors V and W in R 3. | 1 0 1 | | 1 | | 0 | False. T = | 0 1 1 | V = | 0 | W = | 0 | | 0 0 1 | | 0 | | 1 | | 0 | | 0 | | 1 | | 1 | | 0 | T[VxW] = T| -1 | = |-1 | (TV)x(TW) = | 0 | x| 1 } = | -1 |. | 0 | | 0 | | 0 | } 1 } | 1 |

51. 49. There is an invertible 10x10 matrix that has 92 ones among its entries. False. There are only 8 entries which are not one. At least 2 columns have only ones. Matrices with 2 identical columns are not invertible.

52. 50. The formula rref(AB) = rref(A)rref(B) holds for all mxn matrices A and for all nxp matrices B. False A = B = | 0 0 | | 1 0 | rref(AB) =| 0 0 | rref(A)rref(B) = | 1 0 | | 0 0 | | 0 0 |