Matrices With Applications In Statistics 2nd Edition Franklin A Graybill - Solutions

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What is the value of Σ(X – 2) for the following scores: 6, 2, 4, 2?a. 12b. 10c. 8d. 6
What value is represented by the lowercase letter n?a. the number of scores in a populationb. the number of scores in a samplec. the number of values to be added in a summation problemd. the number of steps in a summation problem
Identify what is represented by each of the following symbols: X, Y, N, n, and ∑.
The teacher in a communications class asks students to identify their favorite reality television show. The different television shows make up a scale of measurement.a. nominalb. ordinalc. intervald. ratio
When measuring height to the nearest half inch, what are the real limits for a score of 68.0 inches?a. 67 and 69b. 67.5 and 68.5c. 67.75 and 68.75d. 67.75 and 68.25
A researcher studies the factors that determine the number of children that couples decide to have. The variable, number of children, is an example of a variable.a. discreteb. continuousc. nominald. ordinal
An operational definition is used to a hypothetical construct.a. defineb. measurec. measure and defined. None of the other choices is correct.
Differentiate nominal, ordinal, interval, and ratio scales of measurement.
Explain why operational definitions are developed for constructs and identify the two components of an operational definition.
Stephens, Atkins, and Kingston (2009) found that participants were able to tolerate more pain when they shouted their favorite swear words over and over than when they shouted neutral words. For this study, what is the independent variable?a. the amount of pain toleratedb. the participants who
A research study comparing alcohol use for college students in the United States and Canada reports that more Canadian students drink but American students drink more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). What research design did this study use?a. correlationalb. experimentalc.
In a correlational study, how many variables are measured for each individual and how many groups of scores are obtained?a. 1 variable and 1 groupb. 1 variable and 2 groupsc. 2 variables and 1 groupd. 2 variables and 2 groups
Differentiate correlational, experimental, and nonexperimental research and describe the data structures associated with each.
What statistical concept explains the difference between your mean and the mean for the entire group?a. statistical errorb. inferential errorc. descriptive errord. sampling error
IQ tests are standardized so that the average score is 100 for the entire group of people who take the test each year. However, if you selected a group of 20 people who took the test and computed their average IQ score you probably would not get
In general, statistical techniques are used to summarize the data from a research study and statistical techniques are used to determine what conclusions are justified by the results.a. inferential, descriptiveb. descriptive, inferentialc. sample, populationd. population, sample
Statistical techniques that summarize, organize, and simplify data are classified as .a. population statisticsb. sample statisticsc. descriptive statisticsd. inferential statistics
A researcher is curious about the average IQ of registered voters in the state of Florida.The entire group of registered voters in the state is an example of a .a. sampleb. statisticc. populationd. parameter
A researcher is interested in the sleeping habits of American college students.A group of 50 students is interviewed and the researcher finds that these students sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an example of a(n) .a. parameterb. statisticc.
Describe the concept of sampling error and explain how this concept creates the fundamental problem that inferential statistics must address.
Define descriptive and inferential statistics and describe how these two general categories of statistics are used in a typical research study.
Define the terms population, sample, parameter, and statistic, and describe the relationships between them.
41. In Prob. 40, let B = C-1 and partition B as$$B =\begin{bmatrix}B_{11} & b_{12} \\b_{21} & b\end{bmatrix}$$where B11 is an *n x n* submatrix. Show that(1) b = 0,(2) b12 = (1/n)1,(3) 1'B11 = 0,(4) AB11 and B11A are idempotent.
40. Show that C is nonsingular where C is defined by$$C =\begin{bmatrix}A & 1 \\1' & 0\end{bmatrix}$$and A is defined in Prob. 38.
39. Show that Ak + J is nonsingular where A is defined in Prob. 38 and *k* is any positive integer.
38. Let A be a symmetric *n x n* matrix of rank *n - 1* such that **1'A = 0**; that is, every column of A adds to zero. Show that **B = A + (1/n)11'** is nonsingular, and the inverse is A + (1/n)J.
37. If A is an *m x n* matrix and if the rank of A is *n*, show that A-1 is unique.
36. Consider the linear model y Xẞe and the normal equations
35. Prove Theorem 7.6.6.=A(A) for any L-inverse of A.be any L-inverse of A'A. Show that
34. Let A be any matrix and let (A'A)A'A(A'A).
33. If A is a symmetric matrix show that A
32. In Prob. 31 show that B = AA.
31. Prove that B is symmetric idempotent where BA(A'A) A' and where (A'A)is any c-inverse of A'A.
30. Prove Theorems 6.6.1, 6.6.2, 6.6.3, 6.6.7, and 6.6.8 if the c-inverses are re-placed by L-inverses.
29. In Prob. 28 show that B is a symmetric c-inverse of A where B =$$\frac{1}{2}$$ [A + (A)']and where A is any c-inverse of A.
28. If A is symmetric show that for any c-inverse A of A the matrix (A')' is also a c-inverse of A.
27. If A is a symmetric matrix and A is any c-inverse of A, show that A is not necessarily symmetric.
26. In Prob. 25 show that B may not always be a g-inverse of A, since BA may not always be symmetric.
25. Let A be any mn matrix and define B by B A¹.(A'A) A' where (A'A) is any e-inverse of A'A. Show that ABAA; BABB; AB is symmetric.
24. Prove that if A is nonsingular, then A A A
23. Prove that if AA I then AAI for all L-inverses of A.
22. Prove that for any matrix A if there is an L-inverse A¹ such that AA² = I then AAI.
21. Find an L-inverse of the matrix A where A =$$\begin{bmatrix}1 & 2 & 7 \\1 & 1 & 5 \\-1 & 2 & 1 \\2 & 1 & 8\end{bmatrix}$$.
20. For the system X'Xß = X'y in Sec. 7.5, prove that Gß is unique for any solutionß if and only if the column space of G' is a subspace of the column space of X'.
19. In Prob. 18 find a 2 x 3 matrix G such that Gx is unique for any solution vector x.
18. Show that the system of equations below is consistent and that a unique solution does not exist.4x1 + 2x2 + 2x3 = 3, 2x1 + 2x2 = 0, 2x1 + 2x3 = 3.
17. In Prob. 16 find a least squares solution by finding an L-inverse of the matrix.
18. Show that the system of equations below is consistent and that a unique solution does not exist.$$\begin{aligned}4x_1 + 2x_2 + 2x_3 &= 3,\\2x_1 + 2x_2 &= 0,\end{aligned}$$--- OCR End ---
17. In Prob. 16 find a least squares solution by finding an L-inverse of the matrix.
16. Show that the system below is inconsistent.$$\begin{aligned}x_1 + x_2 + x_4 &= 1,\\x_1 + x_2 + x_5 &= 2,\\x_1 + x_3 + x_4 &= 1,\\x_1 + x_3 + x_5 &= 3.\end{aligned}$$
15. In Prob. 13 find an LSS.
14. In Prob. 13 find the BAS.
13. Show that the system below is inconsistent.$$\begin{aligned}x_1 + x_2 + x_3 &= 3,\\x_1 - x_2 + 2x_3 &= -3,\\3x_1 - x_2 + 5x_3 &= -2,\\2x_1 - x_2 - x_3 &= 4.\end{aligned}$$
12. Let x₁ and x₂ be any solutions to Ax = g. Show that the vector x₁ - x₂ is orthogonal to the rows of A.
11. Let x₀ be a solution to Ax = g where x₀ = A⁻¹g. Show that y is a solution where y = x₀ + z for all vectors z that belong to the orthogonal complement of the column space of A'.
10. Prove that for any matrix A if AA⁻¹ = I, then AA⁻¹ = I for each c-inverse of A.
9. Prove that for any matrix A if there is a c-inverse AC, such that AAc = I, then AA-=1.
8. If x1, x2,..., xr are solutions to the system Ax = g, show that y = ∑i=1rcixi--- OCR End ---
7. Consider the system Ax = g and any c-inverse Ac of the m x n matrix A. Let h1, h2,..., hr be any set of vectors from En, and let c1, c2,..., cr be any set of scalars. Show that if the system is consistent the vector y is a solution to the system where y = Acg + ∑i=1rci(I - AcA)hi.
6. Find any two distinct solutions x1 and x2 to the system in Prob. 1 and demonstrate that $$ \frac{1}{2}x_1 + \frac{1}{2}x_2$$ is also a solution.
5. In Prob. I find a linearly independent set of solution vectors.
4. In Prob. I find the number of linearly independent solution vectors.
3. In Prob. 1 show that the system is consistent by using Theorem 7.2.3.
2. In Prob. 1 find Ac, a conditional inverse of A.
1. Show that the system of equations Ax = g below is consistent by using Theorem 7.2.2.x1 - 2x2 + 3x3 - 2x4 = 2 x1 + x3 - 3x4 = -4 x1 + 2x2 - 3x3 = -4
76. Let A be any n x n matrix and let H be its Hermite form. Show that A-A = HH.
75. Let A be an m x n matrix and B be an n x m matrix. Show that if ABB- = A and B'B- = AB, then A = B-.
74. If A, B, and X are m x n, k x n, and m x k matrices, respectively, show that XX- = AA- if A = XB and X = AC.
73. Let A be an m x n matrix of rank m and let B be an m x m matrix of rank m.Show that (A'BA)- = A B' A'.
72. Let A be an m x n matrix. Show that A- = A'(AA')A(A'A)'A', where (A'A)' and (AA')' are any c-inverses of the respective matrices.
71. Let A be an m x n matrix and let B be an n x p matrix. Show that (AB)- = B-A- if either (1) or (2) below is true.(1) A'A = I.(2) BB' = I.
70. Let A be an *m* x *n* matrix and let B be an *n* x *p* matrix of rank *n*. Show that(AB)(AB) = AA.
69. If a is an *m* x 1 vector and b is an *n* x 1 vector, find a c-inverse of ab' in terms of a and b.
68. If ABA = kA for k ≠ 0, show that (1/k)B is a c-inverse of A.
67. If A' is a c-inverse of A, show that B is also a c-inverse of A, where B =A'AA' + (I - A'A)P + Q(I - AA'), where P and Q are any matrices of appropriate sizes.
66. Let A be an m x n matrix and let B be an n x k matrix. Show that (AB)' =BCA (where (AB), A, B' are any c-inverses of the respective matrices) if and only if A'AB'B is idempotent.
65. Let A be an n x n symmetric matrix such that A² = mA. Show that B =1/m A is a g-inverse of A.
64. Let A ≠ 0 be an m x n matrix. Show that there exist matrices B and C such that BA°C = I, where A' is any c-inverse of A.
63. Let A be any symmetric n x n matrix. Show that B is a symmetric c-inverse of A, where B = ½[A + (A')'], where A' is any c-inverse of A.
62. Show that A'AB = 0 if and only if AB = 0 for any matrices A and B such that the multiplications are defined.
61. For A and B in Prob. 56, show that F is nonsingular, where$$F = \begin{bmatrix} A+ BB' & B \\ B' & 0 \end{bmatrix}$$
60. If A and B are defined as in Prob. 56, show that C is nonsingular, where$$C = \begin{bmatrix} A & B \\ B' & 0 \end{bmatrix}$$and show that$$C^{-1} = \begin{bmatrix} A^{-1} & B' \\ B^{-1} & 0 \end{bmatrix}$$
59. In Prob. 56, show that A+ BB' is nonsingular and (A + BB')-1 =A' + BB'.
58. In Prob. 56, show that AA BB = I.
57. Let A and B be as given in Prob. 56. Show that A+ BB' is nonsingular.
56. If A is an m x m symmetric matrix of lank k < rn, show that there exists an m x m - k matrix B of rank rn - k such that B'A = O.
55. Let A be an rn x n matrix. Show that B is an orthogonal left identity fer A where B =2AA- - I.
54. Prove Theorem 6.7.10.
53. Prove Theorem 6.7.1.
52. In Prob. 6, find the Hermite form of BA where$$B =\begin{bmatrix}1 & 2 & 1 \\1 & -1 & -1 \\1 & -1 & 1\end{bmatrix},$$and show that it is the same as the Hermite form of A.
51. In Prob. 6, show by using the Hermite form of A that the first two columns of A are linearly independent and find the linear combination of these two columns that is equal to the third column.
50. Find the Hermite form of the matrix A in Prob. 6.
49. In Prob. 47, let C = BAB. Show that CAC = C.
48. In Prob. 47, show that BAB has the same rank as A.
47. If B is any c-inverse of A, show that BAB is also a c-inverse of A.
46. Let P and Q be respectively m × m and n × n nonsingular matrices and let A be any m × n matrix. Show that there exists a c-inverse of PAQ denoted by (PAQ)ºsuch that (PAQ)º = Q¯¹AºP¯¹ where Aº is any c-inverse of A.
45. If Aº is any c-inverse of a matrix A, show that (Aº)' is a c-inverse of A'.
44. Show that a c-inverse of a singular diagonal matrix is not unique.
43. If A is defined by$$A =\begin{bmatrix}B & 0 \\0 & C\end{bmatrix},$$show that Aº is a c-inverse of A where$$Aº =\begin{bmatrix}Bº & 0 \\0 & Cº\end{bmatrix},$$where Bº and Cº are any c-inverses of B and C, respectively.
42. If A is nonsingular, show that a c-inverse of A is unique and Aº = A¯¹.

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Matrices With Applications In Statistics 2nd Edition Franklin A Graybill - Solutions

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