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

61. For A and B in Prob. 56, show that F is nonsingular, where$$F = [\begin{matrix}A + BB' & B \\B' & 0\end{matrix}]$$.
60. If A and B are defined as in Prob. 56, show that C is nonsingular, where C = [A B B' 0]and show that$$C^{-1} = [\begin{matrix}A & B \\B' & 0\end{matrix}]^{-1}$$.
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$$B = \begin{bmatrix} 1 & 2 \\\ 1 & -1 \\\ -1 & -1 \end{bmatrix}$$, 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 x m and n x n nonsingular matrices and let A be any m x n matrix. Show that there exists a c-inverse of PAQ denoted by (PAQ)'such that (PAQ)' = Q'A'P-1 where A' is any c-inverse of A.
45. If A' is any c-inverse of a matrix A, show that (A')' 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 A' is a c-inverse of A where$$A' = \begin{bmatrix} B' & 0 \\\ 0 & C' \end{bmatrix}$$, where B' and C' are any c-inverses of B and C, respectively.
42. If A is nonsingular, show that a c-inverse of A is unique and A' = A-1.
41. Does there ever exist a c-inverse of A' 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.
38. For the matrix$$A = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}$$
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 A and A be any two c-inverses of the *m x m* matrix A, and let g be any*n x 1* vector such that AA'g = g. Show that AA'g = g.
34. Let A be an *m x m* 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 *m x n* matrix of rank *m* such that A = BC where B and C each has rank *m*. Show that (BC)' = C'B'.
31. Let A be an *m x n* matrix and let B be an *n x k* 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 *k = n* and P is nonsingular, show that P'A' is a c-inverse of AP.
28. Let A be an *m x n* matrix and let P be any *n x k* matrix of rank *n*. Show that P'A' is a c-inverse of AP, where A' and P' are any c-inverses of A and P, respectively.
27. If A is an *m x n* matrix, B is an *m x n* matrix and AB' = 0, and B'A = 0, show that(1) A'B = 0,(2) B'A' = 0,(3) AB' = 0,(4) BA' = 0,(5) B'A = 0,(6) A'B' = 0.
25. If A is an m x m symmetric matrix such that a'A = 0, show that 'A = 0(a is an m x 1 vector).
24. Let λi (i = 1, 2, ..., r) be the nonzero characteristic roots of an m x m symmetric matrix A. Show that λi-1 (i = 1, 2, ..., r) are the nonzero characteristic roots of A-1.
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 diagonal matrix.
22. Prove Theorem 6.3.3.
21. If A is a given m x n 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.
19. Let A be an m x n given matrix, and let X be any n x m matrix such that A'AX = A'is satisfied, and let Y be any n x m matrix such that YAA' = A'is satisfied. Show that the g-inverse of A is given by AYAX.
18. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & -1 & 1 \\2 & -2 & 0 \\3 & 3 & -3\end{bmatrix}$$
17. Find the g-inverse of the matrix A where--- OCR End ---
16. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 & 0 \\0 & 0 & 0 & 2 & 2 \\0 & 0 & 0 & 2 & 2\end{bmatrix}$$
15. Find the g-inverse of the symmetric matrix A where$$A = \begin{bmatrix}3 & 1 & 0 & 1 \\1 & 4 & -1 & 2 \\0 & -1 & 6 & 2 \\1 & 2 & 2 & 4\end{bmatrix}$$
14. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}6 & 1 & 2 & 4 & 9 \\-3 & 1 & 5 & 2 & 7 \\1 & 0 & 3 & -4 & 1 \\1 & 3 & 17 & 4 & 24 \\1 & -1 & -13 & 6 & -9\end{bmatrix}$$
13. Prove the following: Let A be an m x n matrix, X be an n x r matrix, C be an m x r matrix, B be an r x g matrix, and D be an n x g matrix. A necessary and sufficient condition that the two equations AX = C and XB = D have a common solution X is (1) each equation has a solution and (2) AD = CB.
12. Let the m x 2 (m≥2) matrix A of rank 1 be defined by A = [a, ca]where c is a scalar and a is an m x 1 nonzero vector. Find the g-inverse of A in terms of a and c.
10. Show that the system of equations given below is not consistent.6x1 + x2 - 3x3 + x4 = 0, 4x1 - x2 + x3 - 2x4 = 5,
9. Find the general solution to the system of equations in Prob. 7.
8. Find a solution to the system of equations in Prob. 7.
7. Use Theorem 6.3.1 to show that the system of equations given below is consistent.3x1 - 2x2 + x3 = 3, 3x1 + x2 + 2x3 = 5, 3x1 + 10x2 + 5x3 = 11.
6. Find the g-inverse of the 3 x 3 matrix A where$$A =\begin{bmatrix}3 & 2 & 1 \\1 & 1 & 1 \\3 & 1 & -1\end{bmatrix}$$using the methods presented in each of the Theorems 6.5.1, 6.5.2, 6.5.5, and 6.5.8.
5. Show that the g-inverse of a general 2 x 2 symmetric matrix A of rank 1 defined by A =$$\begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{bmatrix}$$is given by$$A^- =\begin{bmatrix}\frac{a_{11}}{T} & \frac{a_{12}}{T} \\\frac{a_{21}}{T} & \frac{a_{22}}{T}\end{bmatrix}$$where T = a11 + a12 +
4. Find the g-inverse of the 5 x 2 matrix A where$$A = \begin{bmatrix} 2 & 4\\ 1 & 2 \\ 3 & 6 \\ 5 & 10 \\ 2 & 4 \end{bmatrix}$$Use Theorem 6.5.1.
3. Find the g-inverse of the 6 x 2 matrix A where$$A = \begin{bmatrix} 1 & 1\\ 3 & 3 \\ 5 & 2 \\ 2 & 1 \\ 0 & 6 \\ 1 & 5 \end{bmatrix}$$Use Theorem 6.2.16.
2. Find the g-inverse of the 2 x 2 matrix A where$$A = \begin{bmatrix} -1 & -1\\-1 & -1 \end{bmatrix}.$$Use Theorems 6.4.1 and 6.4.9.
1. Find the g-inverse of the vector a where$$\mathbf{a} = \begin{bmatrix} 1\\3\\1\\5\\2 \end{bmatrix}$$Use Theorem 6.4.8.
23. By the method in Prob. 22, find the inverse of I - A where$$A = \begin{bmatrix}0 & 1 & -1 \\0 & 0 & 2 \\0 & 0 & 0\end{bmatrix}.$$
22. In Prob. 21, show (I - A)-1 = Σi=0k-1 Ai.
21. Suppose A is an n x n matrix and Ak = 0 for a positive integer k. Show that I - A is nonsingular.
20. Prove Theorem 5.6.4.
19. Prove Theorem 5.6.2.
18. Prove Corollary 5.3.3.
17. In Prob. 16, find a matrix C such that AC = B.
16. Let A and B be defined by$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3\end{bmatrix}; B = \begin{bmatrix}0 & 2 & 1 \\0 & 4 & 2 \\0 & -2 & -1\end{bmatrix}.$$Show that the column space of B is a subspace of the column space of A.
15. Prove Theorem 5.4.5.
14. Let A and B be m x n matrices of rank n where n < m. Show that A and B may not have the same column space.
13. If A and B are nonsingular n x n matrices, prove that they have the same column space.
12. Find the dimension of the column space of A where$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3 \\1 & 2 & 0\end{bmatrix}$$
11. In Prob. 10, show that z₁ is the projection of y into S by using the results of Theorem 4.4.1.
10. Let S be spanned by x₁ and x₂, where x₁ = [1, 1, 0, -1]; x₂ = [0, 1, 1, 1]; the
9. Find the vector z and the scalar λ such that y = λx + z where$$y = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}; x = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}$$and such that x and z are orthogonal.
8. Find a basis for the orthogonal complement of the subspace in E₃ spanned by x where x' = [1, 1, -1].
7. Prove Theorem 5.2.5.
6. Use the results of Probs. 4 and 5 to demonstrate Theorem 5.2.5.
5. In Prob. 1, find the dimension of S₁ ⊕ S₂.
4. In Prob. 1, find the dimension of S₁ ∩ S₂.
3. If {a₁, ..., a,} spans S₁ and {b₁, ..., b,} spans S₂, prove that {a₁, ...,a, b₁, ..., b,} spans S₁ ⊕ S₂.
2. In Prob. 1, find a basis for S₁ ⊕ S₂.
1. If a₁, a₂, a₃ span S₁ and b₁, b₂, b₃ span S₂, find a basis for S₁ ∩ S₂.$$a₁ = \begin{bmatrix} 1 \\ 3 \\ 2 \\ -1 \end{bmatrix}; a₂ = \begin{bmatrix} 0 \\ -1 \\ 2 \\ 1 \end{bmatrix}; a₃ = \begin{bmatrix} 2 \\ 7 \\ 2 \\ -3 \end{bmatrix};$$$$b₁ = \begin{bmatrix} 1 \\ 8
8. Find a solution to the system of equations in Prob. 7.
9. Find the general solution to the system of equations in Prob. 7.
10. Show that the system of equations given below is not consistent.6x1 + x2 - 3x3 + x4 = 0, 4x1 - x2 + x3 - 2x4 = 5,--- OCR End ---
11. In Prob, 10,show that the first three equations are consistent.
12. Let the m x 2 (m≥2) matrix A of rank 1 be defined by A = [a, ca]where c is a scalar and a is an m x 1 nonzero vector. Find the g-inverse of A in terms of a and c.
13. Prove the following: Let A be an m x n matrix, X be an n x r matrix, C be an m x r matrix, B be an r x g matrix, and D be an n x g matrix. A necessary and sufficient condition that the two equations AX = C and XB = D have a common solution X is (1) each equation has a solution and (2) AD = CB.
14. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}6 & 1 & 2 & 4 & 9\\-3 & 1 & 5 & 2 & 7\\1 & 0 & 3 & -4 & 1\\1 & 3 & 17 & 4 & 24\\1 & -1 & -13 & 6 & -9\end{bmatrix}$$
15. Find the g-inverse of the symmetric matrix A where$$A = \begin{bmatrix}3 & 1 & 0 & 1\\1 & 4 & -1 & 2\\0 & -1 & 6 & 2\\1 & 2 & 2 & 4\end{bmatrix}$$
7. Use Theorem 6.3.1 to show that the system of equations given below is consistent.3x1 - 2x2 + x3 = 3, 3x1 + x2 + 2x3 = 5, 3x1 + 10x2 + 5x3 = 11.
6. Find the g-inverse of the 3 x 3 matrix A where$$A = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & -1 \\ 3 & 1 & -1 \end{bmatrix}$$using the methods presented in each of the Theorems 6.5.1, 6.5.2, 6.5.5, and 6.5.8.
21. Suppose A is an n x n matrix and A^k = 0 for a positive integer k. Show that I - A is nonsingular.
22. In Prob. 21, show (I - A)^-1 = Σ^(k-1)_i=0 A^i.
23. By the method in Prob. 22, find the inverse of I - A where$$A = \begin{bmatrix}0 & 1 & -1 \\0 & 0 & 2 \\0 & 0 & 0\end{bmatrix}$$
1. Find the *g*-inverse of the vector *a* where 13*a* = 1 52 Use Theorem 6.4.8.
2. Find the *g*-inverse of the 2 x 2 matrix *A* where-1 -1*A* =-1 -1 Use Theorems 6.4.1 and 6.4.9.
3. Find the *g*-inverse of the 6 x 2 matrix *A* where 1 1 3 3 5 2*A* =2 1 0 6 1 5 Use Theorem 6.2.16.
4. Find the *g*-inverse of the 5 x 2 matrix *A* where
5. Show that the g-inverse of a general 2 x 2 symmetric matrix A of rank 1 defined by$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$is given by$$A = \begin{bmatrix} a_{11} & a_{12} \\ T & T \\ a_{21} & a_{22} \\ T & T \end{bmatrix}$$where T = a11 + a12 + a21 + a22 = tr
16. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\0 & 0 & 0 & 2 & 2\\0 & 0 & 0 & 2 & 2\end{bmatrix}$$

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