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Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Office 432 Carver Phone 515-294-8141 E-mail: [email protected]PowerPoint Presentation

Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Office 432 Carver Phone 515-294-8141 E-mail: [email protected]

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Spring, 2003

Hentzel

Time: 1:10-2:00 MWF

Room: 1324 Howe Hall

Office 432 Carver

Phone 515-294-8141

E-mail: [email protected]

http://www.math.iastate.edu/hentzel/class.307.ICN

Text: Linear Algebra With Applications,

Second Edition Otto Bretscher

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.

1. The function T|x| = |x-y| is a

|y| |y-x|

linear transformation.

True. It has matrix | 1 -1 |.

|-1 1 |

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

- 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.

- 4. The formula (A2)-1 = (A-1)2 holds
- for all invertible matrices A.
- True. A A A-1 A-1 = I.

- 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 |

- 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.

- 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.

- 8. The function T|x| = |y| is a linear be a 5x3 matrix.
- |y| |1|
- transformation.
- False. T (2 |0|) = |0| =/= 2 T|0| = | 0 |
- |0| |1| |0| | 2 |

- 9. The matrix | 5 6 | represents a be a 5x3 matrix.
- |-6 5 |
- rotation-dilation.
- True. The dilation is by Sqrt[61] the angle
- is ArcTan[-6/5] = -0.876058 radians

- 10. If A is any invertible nxn matrix, then be a 5x3 matrix.
- A commutes with A-1.
- True. By definition, A A-1 = A-1A = I

- 11. Matrix | 1 2 | is invertible. be a 5x3 matrix.
- | 3 6 |
- False. The RCF is | 1 2 |.
- | 0 0 |

- | 1 1 1 | be a 5x3 matrix.
- 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 |

- 13. There is an upper triangular 2x2 be a 5x3 matrix.
- matrix A such that A2 = | 1 1 |
- | 0 1 |
- True. A = | 1 1/2 | is one possibility.
- | 0 1 |

- 14. The function be a 5x3 matrix.
- 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|

- 15. Matrix | k -2 | is invertible for all be a 5x3 matrix.
- | 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.

- 16. There is a real number k such that the be a 5x3 matrix.
- matrix | k-1 -2 | fails to be invertible.
- | -4 k-3 |
- True. k = -1 | -2 -2 | k = 5 | 4 -2 |.
- | -4 -4 | | -4 2 |

- 17. There is a real number k such that be a 5x3 matrix.
- 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.

- 18. Matrix | -0.6 0.8 | represents a be a 5x3 matrix.
- |-0.8 -0.6 |
- rotation.
- True: theta = Pi + ArcCos[0.6] = 4.06889

- 19. The formula det(2A) = 2 det(A) holds be a 5x3 matrix.
- for all 2x2 matrices A.
- False. det(2A) = 4 det(A).

- 20. There is a matrix A such that be a 5x3 matrix.
- | 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 |

- 21. There is a matrix A such that be a 5x3 matrix.
- 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 |

- 22. There is a matrix A such that be a 5x3 matrix.
- | 1 2 | A = | 1 1 |,
- | 1 2 | | 1 1 |
- True. | 1 1 | works.
- | 0 0 |

- 23. Matrix | -1 2 | represents a shear. be a 5x3 matrix.
- | -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 |

- 24. | 1 k | be a 5x3 matrix.3 = | 1 3k | for all real
- | 0 1 | | 0 1 |
- numbers k.
- True:

- 25. The matrix product be a 5x3 matrix.
- | a b | | d -b | is always a scalar
- | c d | | -c a |
- of I2.
- True. The scalar is ad-bc.

- 26. There is a nonzero upper triangular be a 5x3 matrix.
- 2x2 matrix A such that A2 = | 0 0 |.
- | 0 0 |
- True. A = | 0 1 | is one possibility.
- | 0 0 |

- 27. There is a positive integer n such that be a 5x3 matrix.
- | 0 -1 | n = I2.
- | 1 0 |
- True. n = 4 is one possibility.

- 28. There is an invertible 2x2 matrix A be a 5x3 matrix.
- 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 |

- 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.

- 30. If A rows.2 = In, then matrix A must be invertible.
- True. In fact, A is its own inverse.

- 31. If A rows.17 = I2, then A must be I2.
- False A = | Cos[t] -Sin[t] |
- | Sin[t] Cos[t] |
- Where t = 2 Pi/17 should work.

- 32. If A rows.2 = I2 , then A must be either I2 or –I2.
- False A = | -1 0 | is one possibility.
- | 0 1 |

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

- 34. If A and B are two 4x3 matrices such rows.
- that AV = BV for all vectors v in R3, then
- matrices A and B must be equal.
- True. It follows that AI = BI for the 3x3
- identity matrix I. Thus A=B.

- 35. If matrices A and B commute, then the rows.
- formula A2B = BA2 must hold.
- True. A2B = AAB = ABA=BAA=BA2.

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

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

- 38. The equation A rows.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.

- 39. If matrix | a b c | is invertible, then rows.
- | 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 |

- 40. If A rows.2 is invertible, then
- matrix A itself must be invertible.
- True. For A2 to be defined, then
- A must be square. If AAB = I, then
- A must be right invertible so A is
- invertible.

- 41. The equation A rows.-1 = A holds for all 2x2
- matrices A representing a reflection.
- True. For a reflection A2 = I.

- 42. The formula (AV).(AW) = V.W holds rows.
- for all invertible 2x2 matrices A and for
- all vectors V and W in R2.
- False. | 1 1 | | 0 | .| 1 1 | | 1 | = 1
- | 0 1 | | 1 | | 0 1| | 0 |

- 43. There exist a 2x3 matrix A and a 3x2 rows.
- matrix B such that AB = I3.
- True. | 1 0 0 | | 1 0 | = | 1 0 |
- | 0 1 0 | | 0 1| | 0 1 |
- | 0 0|

- 44. There exist a 3x2 matrix A and a 2x3 rows.
- matrix B such that AB = I3.
- False. There must be some X =/= 0
- such that BX = 0. Then 0 = ABX = X.
- Contradiction.

- 45. If A rows.2 + 3A + 4 I3 = 0 for a 3x3 matrix
- A then A must be invertible.
- True. A(A+3) = -4 I3
- so the inverse of A is (-1/4)(A+3).

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

- 47. If matrix A represents a shear, then rows.
- the formula A2-2A+I2 = 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.

- 48. If T is any linear transformation rows.
- from R3 to R3, then T(VxW) = T(V)xT(W)
- for all vectors V and W in R3.
- | 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 |

- 49. There is an invertible 10x10 matrix rows.
- 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.

- 50. The formula rref(AB) = rref(A)rref(B) rows.
- 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 |

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