January 22 Review questions Math 307 Spring 2003

January 22 Review questions

• • • 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

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.

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.

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

3. There is a 3 x 4 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.

4. If A is a 3 x 4 matrix and vector V is in R^4, then vector AV is in R^3. True: A V 3 x 4 = (AV) 4 x 1 3 x 1

5. If the 4 x 4 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.

6. 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.

7. There is a 5 x 5 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 any solutions at, there are infinitely many of them.

8. 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_|

9. 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.

10. If vector U is a linear combination of vectors V and W, then we can write U = a. V+b. W 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.

11. The rank of the following matrix is 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 |

12. | 11 13 15| | -1 | | 17 19 21| | 3 | | -1 | | 13 | = | 19 | | 21 | False. It cannot possibly be correct since a 2 x 3 matrix times a 3 x 1 matrix will be a 2 x 1 matrix, not a 3 x 1 matrix.

13. 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.

14. |1| Vector | 2 | is a linear combination of |3| |4| Vectors | 5 | |6| True: and |7| |8|. |9| |4| |7| 2|5| -1 |8| |6| |9| = |1| |2| |3|

15. 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 0 x + 0 y + 0 z = 3 which cannot possible be true.

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

17. 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 | | 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.

18. The rank of this matrix is 3. | 1 1 1| | 1 2 3| | 1 3 6| True: | 1 1 1 | | 1 0 -1 | | 1 0 0 | | 1 2 3|~| 0 1 2|~|0 1 0| | 1 3 6| | 0 2 5| |00 1| | 0 01|

19. The system is inconsistent for any (4 x 3) matrix A. | 0 | AX=| 0 | | 1 | False: | | 0 0 0 1 0 0 0||1| 0| |0| = |0| |1|

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

21. 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 |~| 0 1 a | |-a -b 0 | | -a –b 0 | | 0 –b –ab | | 0 0 0 | The Row Canonical Form has exactly 2 non zero rows.

22. 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.

23. 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 | | 0| |0| W=|0| | 1|

24. 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.

25. If A and B are any two 3 x 3 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 3 x 3 matrices in Row Canonical Form. | 1 0 0| |0 1 0| |0 0 1| | 0 0 0| |0 0 0|

26. 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 = a. V + b W and V = c. P+d. Q+e. R, then U = a(c. P+d. Q+e. R)+b. W = ac. P+ad. Q+ae. R+b. W.

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

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

29. If the system AX = B has a unique solution then A must be a square matrix. 30. False. | 1 0 0 | | x | | 1 | | 010||y|=|1| | 001||z| |1| | 111| |3| 31. This has a unique solution and A is not square. 32.

30. If A is a 4 x 3 matrix, then there exists a vector B in R^4 31. such that the system AX = B is inconsistent. 32. True. The system 33. | 1 | 0 0 | 0| 0 | 0| 1 | 0| 0 | 1| 34. is inconsistent. If we use inverse elementary row 35. operations, we can backwards transform this system to 36. any system [ A | B ] where A is a 4 x 3 matrix with rank 3. 37. The backwards transformed system will still be 38. inconsistent.

31. If A is a 4 x 3 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. 32. True: If A V = A W, then A(V-W) = 0. Since A has rank three, 33. the only solution to AX = 0 is X = 0. Thus V-W = 0 and V = W. .

32. If A is a 4 x 4 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.

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 | 0 |+0 | 1| | 0| v w | 2 | = | 0 | u 2 V+ 0 W = U, but W is not a linear combination of U and V.

| 1 0 2| 34 If A = [ U V W ] and RREF(A) = | 0 1 3 | | 0 0 0| then the equation W = 2 U + 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.

35. If A and B are matrices of the same size, then the formula rank(A+B) = rank(A) + rank(B) must hold. 36. False 37. | 1 0 0 | | 0 1 0 | + | 0 0 1 | = | 1 1 0| | 0 1 1| 38. The rank of all three of these matrices is two.

36. If A and B are any two nxn matrices of rank n, then A can be transformed into B by elementary row operations. 37. True. Since A and B are both equivalent to the identity matrix I, we can transform either one of them into the other.