Math 307Spring, 2003

Hentzel

Time: 1:10-2:00 MWFRoom: 1324 Howe Hall

Office 432 CarverPhone 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

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.

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

• |y| |1|

• transformation.

• False. T (2 |0|) = |0| =/= 2 T|0| = | 0 |

• |0| |1| |0| | 2 |

• 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

• 10. If A is any invertible nxn matrix, then

• A commutes with A-1.

• True. By definition, A A-1 = A-1A = I

• 11. Matrix | 1 2 | is invertible.

• | 3 6 |

• False. The RCF is | 1 2 |.

• | 0 0 |

• | 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 |

• 13. There is an upper triangular 2x2

• matrix A such that A2 = | 1 1 |

• | 0 1 |

• True. A = | 1 1/2 | is one possibility.

• | 0 1 |

• 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|

• 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.

• 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 |

• 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.

• 18. Matrix | -0.6 0.8 | represents a

• |-0.8 -0.6 |

• rotation.

• True: theta = Pi + ArcCos[0.6] = 4.06889

• 19. The formula det(2A) = 2 det(A) holds

• for all 2x2 matrices A.

• False. det(2A) = 4 det(A).

• 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 |

• 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 |

• 22. There is a matrix A such that

• | 1 2 | A = | 1 1 |,

• | 1 2 | | 1 1 |

• True. | 1 1 | works.

• | 0 0 |

• 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 |•

• 24. | 1 k |3 = | 1 3k | for all real

• | 0 1 | | 0 1 |

• numbers k.

• True:

• 25. The matrix product

• | 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

• 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

• | 0 -1 | n = I2.

• | 1 0 |

• True. n = 4 is one possibility.

• 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 |

• 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 A2 = In, then matrix A must be invertible.

• True. In fact, A is its own inverse.

• 31. If A17 = 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 A2 = 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

• 5 A is invertible as well.

• True. And (5A)-1 = 1/5 A-1.

• 34. If A and B are two 4x3 matrices such

• 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

• formula A2B = BA2 must hold.

• True. A2B = AAB = ABA=BAA=BA2.

• 36. If A2 = 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,

• then matrix A+B must be invertible as well.

• False. Let B = -A.

• 38. The equation A2 = 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• | 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 A2 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-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

• 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

• 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

• matrix B such that AB = I3.

• False. There must be some X =/= 0

• such that BX = 0. Then 0 = ABX = X.

• Contradiction.

• 45. If A2 + 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 A2 = 0, then

• matrix In+A must be invertible.

• True. (In+A)(In-A) = I.

• 47. If matrix A represents a shear, then• • 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

• 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

• 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)• 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 |