January 22

1. January 22 Review questions

2. Math 307 • Spring 2003 • Hentzel • Time: 1:10-2:00 MWF • Room: 1324 Howe Hall • Instructor: Irvin Roy Hentzel • Office 432 Carver • Phone 515-294-8141 • E-Mail: hentzel@iastate.edu • Office Hours: 9:00-10:00 MWF • 2:00- 3:00 MWF

3. We apply the ideas we have learned • We go over the 36 true false questions so that we learn how to use the facts we have learned so far to get really nice results.

4. 1 The following matrix is in RREF. | 1 2 0 | | 0 0 1 | | 0 0 0 | True. | |_1__ 2_ 0 | | 0 0 |_ 1 _| | 0 0 0 | Find the: (a) Stairs (b) Stair step ones (c) Zeros below the stairs (d) Zeros above the stair step ones.

5. A system of four equations in three unknowns is always inconsistent. False: x + y + z = 3 x = 1 y = 1 z = 1 Is consistent.

6. There is a 3x4 matrix with rank 4. False: A matrix with 3 rows can have at most three stair step ones. Thus the matrix can have rank at most 3.

7. If A is a 3x4 matrix and vector V is in R^4, then vector AV is in R^3. True: A V = (AV) 3x4 4x1 3x1

8. If the 4x4 matrix A has rank 4, then any linear system with coefficient matrix A will have a unique solution. True: The Row Canonical Form of [A | B] will always reduce to [ I | B* ] for some B*. There is no stair step one in the last column so there is an answer. There are no parameters so the answer is unique.

9. There exists a system of three linear equations with three unknowns that has exactly three solutions. False: If there is more than one solution, then there is at least one parameter and there will be an infinite number of solutions.

10. There is a 5x5 matrix A of rank 4 such that the system AX = 0 has only the solution X = 0. False: Rank 4 means four stair step ones. One of the variables will not be above a stair step one. Therefore, there is at least one parameter. There may be no solutions at all, but if there are any solutions at, there are infinitely many of them.

11. If matrix A is in RREF, then at least one of the entries in each column must be 1. False: Consider this matrix in RREF. | |_1_ 0 0 0 | | 0 |_1_ 0_ 0 | | 0 0 0 |_1_|

12. If A is an nxn matrix and X is a vector in R^n, then the product AX is a linear combination of the columns of the matrix A. True: The columns of AB are linear combinations of the columns of A.

13. If vector U is a linear combination of vectors V and W, then we can write U = aV+bW for some scalars a and b. True: This is exactly what we mean when we say that a vector is a linear combination of two vectors.

14. The rank of the following matrix is 2. | 2 2 2 | | 2 2 2 | | 2 2 2 | False: Reduce it to Row Canonical Form and count the non zero rows. The rank is 1. | 1 1 1 | | 0 0 0 | | 0 0 0 |

15. | 11 13 15| | -1 | | 13 | | 17 19 21| | 3 | = | 19 | | -1 | | 21 | False. It cannot possibly be correct since a 2x3 matrix times a 3x1 matrix will be a 2x1 matrix, not a 3x1 matrix.

16. There is a matrix A such that | -1 | | 3 | A | 2 | = | 5 | . | 7 | True: | 0 3/2 | | 0 5/2 | | 0 7/2 | Is such a matrix.

17. | 1 | Vector | 2 | is a linear combination of | 3 | | 4 | | 7 | Vectors | 5 | and | 8 | . | 6 | | 9 | True: | 4 | | 7 | | 1 | 2 | 5 | - 1 | 8 | = | 2 | | 6 | | 9 | | 3 |

18. The system below is inconsistent. | 1 2 3 | | x | | 1 | | 4 5 6 | | y | = | 2 | | 0 0 0 | | z | | 3 | True: The last equation requires that 0x + 0 y + 0 z = 3 which cannot possible be true.

19. There exists a 2x2 matrix A such that A | 1 | = | 3 |. | 2 | | 4 | True: | 1 1 | | 1 | = | 3 | | 2 1 | | 2 | | 4 |

20. If A is a nonzero matrix of the form | a -b | | b a | Then the rank of A must be 2. True: If a = 0, then the RCF = I and the rank is 2. If a =/= 0, then | a -b | ~ | 1 -b/a | ~ | 1 -b | | b a | | b a | | 0 (b^2)/a +a | If (b^2)/a + a = 0, the rank is 1. If (b^2)/a =/= 0, the rank is 2. Since (b^2)/a + a = 0 requires a^2 + b^2 = 0 which cannot happen for real numbers a and b, we know that the rank is always 2.

21. The rank of this matrix is 3. | 1 1 1 | | 1 2 3 | | 1 3 6 | True: | 1 1 1 | | 1 1 1 | | 1 0 -1 | | 1 0 0 | | 1 2 3 | ~ | 0 1 2 | ~ | 0 1 2 | ~ | 0 1 0 | | 1 3 6 | | 0 2 5 | | 0 0 1 | | 0 0 1 |

22. The system is inconsistent for any (4x3) matrix A. | 0 | A X = | 0 | | 0 | | 1 | False: | 0 0 0 | | 1 | | 0 | | 0 0 0 | | 1 | | 0 | | 0 0 0 | | 1 | = | 0 | | 1 0 0 | | 1 |

23. There exists a 2x2 matrix A such that A | 1 | = | 1 | and A | 2 | = | 2 | | 1 | | 2 | | 2 | | 1 | False: A| 2 | = 2 A| 1 | = 2 | 1 | = | 2| =/= | 2 | | 2 | | 1 | | 2 | | 4| | 1 |

24. There exist scalars a and b such that this matrix has rank 3. | 0 1 a || -1 0 b || -a -b 0 | False: | 0 1 a | | -1 0 b | | 1 0 -b | | 1 0 –b | | 1 0 –b || -1 0 b |~| 0 1 a |~| 0 1 a |~| 0 1 a |~| 0 1 a ||-a -b 0 | | -a –b 0 | |-a –b 0 | | 0 –b –ab | | 0 0 0 | The Row Canonical Form has exactly 2 non zero rows.

25. If V and W are vectors in R^4, then V must be a linear combination of V and W. True. V = 1 V + 0 W.

26. If U, V, and W are nonzero vectors in R^2, then W must be a linear combination of U and V. False. U = | 1 | V = | 1 | W = | 0 | | 0 | | 0 | | 1 |

27. If V and W are vectors in R^4, then the zero vector in R^4 must be a linear combination of V and W. True: 0 V + 0 W = 0.

28. If A and B are any two 3x3 matrices of rank 2, then A can be transformed into B by means of elementary row operations. False. Since Row Canonical Forms are unique, we simply display two rank two 3x3 matrices in Row Canonical Form. | 1 0 0 | | 0 1 0 | | 0 1 0 | | 0 0 1 | | 0 0 0 | | 0 0 0 |

29. If vector U is a linear combination of vectors V and W, and V is a linear combination of vectors P, Q and R, then U is a linear combination of P, Q, R, and W. True: U = aV + b W and V = cP+dQ+eR, then U = a(cP+dQ+eR)+bW = acP+adQ+aeR+bW.

30. A system with fewer unknowns than equations must have infinitely many solutions or none. False: x = 1 has exactly one solution y = 1 x+y = 2

31. 28. The rank of any upper triangular matrix is the number of non zeros on the diagonal. False: | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | | 0 0 0 0 | This matrix has rank 3 and no non zeros on the diagonal.

32. If the system AX = B has a unique solution then A must be a square matrix. • False. | 1 0 0 | | x | | 1 | | 0 1 0 | | y | = | 1 | | 0 0 1 | | z | | 1 | | 1 1 1 | | 3 | • This has a unique solution and A is not square.

33. If A is a 4x3 matrix, then there exists a vector B in R^4 • such that the system AX = B is inconsistent. • True. The system • | 1 0 0 | 0 | | 0 1 0 | 0 | | 0 0 1 | 0 | | 0 0 0 | 1 | • is inconsistent. If we use inverse elementary row • operations, we can backwards transform this system to • any system [ A | B ] where A is a 4x3 matrix with rank 3. • The backwards transformed system will still be • inconsistent.

34. If A is a 4x3 matrix of rank 3 and AV = AW for two vectors V and W in R^3, then the vectors V and W must be equal. • True: If A V = A W, then A(V-W) = 0. Since A has rank three, • the only solution to AX = 0 is X = 0. Thus V-W = 0 and V = W..

35. If A is a 4x4 matrix and the system | 2 | A X = | 3 | has a unique solution, | 4 | | 5 | then the system AX = 0 has only the solution X = 0. True. If any system has exactly one solution, then there will be no parameters. Thus every solution is unique.

36. 33. If vector U is a linear combination of vectors V and W, then W must be a linear combination of U and V. False: | 1 | | 0 | | 2 | 2 | 0 | + 0 | 1 | = | 0 | | 0 | | 0 | | 0 | v w u 2V+ 0 W = U, but W is not a linear combination of U and V .

37. | 1 0 2 |34 If A = [ U V W ] and RREF(A) = | 0 1 3 | | 0 0 0 | then the equation W = 2U + 3 V must hold.. | -2 |True. The vector | -3 | must be in the null space of A | 1 | So -2 U -3 V + W = 0. Thus W = 2 U + 3 V.

38. If A and B are matrices of the same size, then the formula rank(A+B) = rank(A) + rank(B) must hold. • False • | 1 0 0 | | 0 1 0 | = | 1 1 0 | | 0 1 0 | + | 0 0 1 | | 0 1 1 | • The rank of all three of these matrices is two.

39. If A and B are any two nxn matrices of rank n, then A can be transformed into B by elementary row operations. • True. Since A and B are both equivalent to the identity matrix I, we can transform either one of them into the other.