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statistics for experimentert

Questions and Answers of Statistics For Experimentert

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)
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
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$$C^{-1} = \begin{bmatrix} A^{-1} & B' \\ B^{-1} & 0
59. In Prob. 56, show that A+ BB' is nonsingular and (A + BB')-1 =A' + BB'.
58. In Prob. 56, show that AA BB = I.
57. Let A and B be as given in Prob. 56. Show that A+ BB' is nonsingular.
56. If A is an m x m symmetric matrix of lank k < rn, show that there exists an m x m - k matrix B of rank rn - k such that B'A = O.
55. Let A be an rn x n matrix. Show that B is an orthogonal left identity fer A where B =2AA- - I.
54. Prove Theorem 6.7.10.
53. Prove Theorem 6.7.1.
52. In Prob. 6, find the Hermite form of BA where$$B =\begin{bmatrix}1 & 2 & 1 \\1 & -1 & -1 \\1 & -1 & 1\end{bmatrix},$$and show that it is the same as the Hermite form of A.
51. In Prob. 6, show by using the Hermite form of A that the first two columns of A are linearly independent and find the linear combination of these two columns that is equal to the third column.
50. Find the Hermite form of the matrix A in Prob. 6.
49. In Prob. 47, let C = BAB. Show that CAC = C.
48. In Prob. 47, show that BAB has the same rank as A.
47. If B is any c-inverse of A, show that BAB is also a c-inverse of A.
46. Let P and Q be respectively m × m and n × n nonsingular matrices and let A be any m × n matrix. Show that there exists a c-inverse of PAQ denoted by (PAQ)ºsuch that (PAQ)º = Q¯¹AºP¯¹
45. If Aº is any c-inverse of a matrix A, show that (Aº)' is a c-inverse of A'.
44. Show that a c-inverse of a singular diagonal matrix is not unique.
43. If A is defined by$$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},$$where Bº and Cº are any
42. If A is nonsingular, show that a c-inverse of A is unique and Aº = A¯¹.
41. Does there ever exist a c-inverse of Aº that is equal to A if A is singular?
40. Show that there does not always exist a c-inverse of a matrix A that equals A.
39. For the matrix in Frob. 15, find a nonsingular matrix B such that BA ... H where H is in Hermite form?
38. For the matrix$$A = \begin{bmatrix}1 & 2 \\1 & 1 \\-1 & 0\end{bmatrix}.$$
37. Show rank (A') ≥ rank (A) for any c-inverse of A. (See Theorem 6.6.8.)
36. Find a c-inverse of the matrix A in Prob. 6.
35. Let A1' and A2' be any two c-inverses of the matrix A, and let g be any vector such that AA1'g = g. Show that AA2'g = g.
34. Let A be an matrix and let P and Q be orthogonal matrices such that PAQ = D where D is a diagonal matrix. Show that QD'P = A'.
33. If A is a positive semidefinite matrix, show that A' is also a positive semidefinite matrix.
32. Let A be an matrix of rank such that A = BC where B and C each has rank . Show that (BC)' = C'B'.
31. Let A be an matrix and let B be an matrix. Define F and G by G = A'AB; F = AGG', and show that AB = FG and (FG)' = G'F'.
30. In Prob. 28, if PP' = I, show that P'A' is a c-inverse of AP.
29. In Prob. 28, if and P is nonsingular, show that P-1A' is a c-inverse of AP.
28. Let A be an matrix and let P be any matrix of rank . Show that PA' is a c-inverse of AP, where A' and P' are any c-inverses of A and P, respectively.
27. If A is an matrix, B is an matrix and AB' = 0, and B'A = 0, show that(1) AB' = 0,(2) B'A = 0,(3) AB = 0.(4) BA' = 0,(5) B'A' = 0.(6) A'B = 0.
26. If A is an I'll x I'll symmetric matrix such that I'A = 0, sbow tha?
25. If A is an mxm symmetric matrix such that a'A=0, show that a'A=0(a is an m x 1 vector).
24. Let λi (i = 1,2,..., r) be the nonzero characteristic roots of an mx m symmetric matrix A. Show that λi (i = 1, 2, ..., r) are the nonzero characteristic roots of A'.
23. Let A be an m x m symmetric matrix and P be an orthogonal matrix such that P'AP = D, where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that P'A P is also a
22. Prove Theorem 6.3.3.
21. If A is a given mxn matrix, find conditions on a matrix X so that the system AX= I is consistent.
20. Find a solution to the system of equations 2x1-x2+x3=8, x1+2x2-x3=-5.

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