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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] 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| |78| |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 = | 1 0| I2. • 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 | • • False. • • | 0 0 1| | 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. • | 101| |1| |0| • False. T = | 0 1 1 | V = | 0 | W = | 0 | • | 001| |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 |