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

3. In Prob. 1 show that the system is consistent by using Theorem 7.2.3.
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.
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
49. In Prob. 47, let C = BAB. Show that CAC = C.
50. Find the Hermite form of the matrix A in Prob. 6.
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.
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.
53. Prove Theorem 6.7.1.
54. Prove Theorem 6.7.10.
55. Let A be an rn x n matrix. Show that B is an orthogonal left identity fer A where B =2AA- - I.
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.
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.
39. For the matrix in Frob. 15, find a nonsingular matrix B such that BA ... H where H is in Hermite form?
40. Show that there does not always exist a c-inverse of a matrix A that equals A.
41. Does there ever exist a c-inverse of Aº that is equal to A if A is singular?
42. If A is nonsingular, show that a c-inverse of A is unique and Aº = A¯¹.
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.
44. Show that a c-inverse of a singular diagonal matrix is not unique.
45. If Aº is any c-inverse of a matrix A, show that (Aº)' is 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.
57. Let A and B be as given in Prob. 56. Show that A+ BB' is nonsingular.
58. In Prob. 56, show that AA BB = I.
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.
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.
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.
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.
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'.
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.
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-.
76. Let A be any n x n matrix and let H be its Hermite form. Show that A-A = HH.
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.
59. In Prob. 56, show that A+ BB' is nonsingular and (A + BB')-1 =A' + BB'.
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$$C^{-1} = \begin{bmatrix} A^{-1} & B' \\ B^{-1} & 0 \end{bmatrix}$$
61. For A and B in Prob. 56, show that F is nonsingular, where$$F = \begin{bmatrix} A+ BB' & B \\ B' & 0 \end{bmatrix}$$
62. Show that A'AB = 0 if and only if AB = 0 for any matrices A and B such that the multiplications are defined.
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.
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.
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.
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.
38. For the matrix$$A = \begin{bmatrix}1 & 2 \\1 & 1 \\-1 & 0\end{bmatrix}.$$
2. In Prob. 1 find Ac, a conditional inverse of A.
20. Consider the plane &'1 through the points ai' 32, 0 wherea, = [I, 1,0,1], 32 = [I, -I, 1,0).Find the projection of the directed line segment t from 0to b onto ei'~ where b' = [0,2, - I, 0]
19. Find the projection of the directed line segment froma, to a2 onto the plane &'~where ei'i goes through 0, xI = [I, 1,I] and X2= [1, -I, I] and where a1 =[1,0, -I], a~ = [2,-I, 1]-
18. In Prob. 16, find the angle between t and fil.
17. In Prob. 16, sketch the three lines.
16. Find the projection of the directed line segment ( froma, to a1 onto the line fil that goes through 0 and b where al = [1,2, -1], a2 ... [I, -1,2), b' = [I, I,ll
15. Let fil be a line through 0 anda, where a' = [I, -1, 1]. Find the equation of the plane r!J that is perpendicular to ~ and passes through the point [2, 1, -1].
14. In Prob. 13, show that this value of d is the minimum distance between 0 and the plane fi'.
13. In Prob. 12, verify that the distance between 0 and tbe point of intersection of the:Iine ~ and the plane f!; is d = (l'(B'B)-ll]-lfl.
12. In Prob. II, find the point on 11 where 2 intersects {?
11. rn Prob. 10,find the equation of a line !l' tbat goes through the origin and is perpendicular to every line in the plane {?
10.Find the equation of the plane {}'that goes through the three points bj = [I, I, 1], bi = [I, -1,0), bJ = [0, 1, -1]. Write the equation in the form a'y = c.
9. Find the equation of the line ~ that is parallel to fI!I and 21, of Prob. 3, equidistant from the two lines, and in the same plane as !i'l and 21.
8. Find the distance from the point x' = [2, 1, -IJ to the line 5£ that goes through the two points a'". [I, 1,0] and b' = [I, -1,2].
7. Find the direction angles of the line through the points a' = [2,0,-I]and b' = [I, -1,3],
6. Find the direction angles of II in Prob, 5.
5. Find the angle between the two directed.line segments II and 12where II is the directed line segment from 0to a' = [1,-I, I] and 12 is the directed line segment from 0 to b' = [2, I, -1].
4. Do the two lines 2, and 21 intersect where fL'1 goes through the two points a' = [I, I, -1]; b'= [2,0,1]and fL' 2 goes through the two points c' = [I, I, I];d' = (2, 0, 2]7
3. Use Def. 4.2.2 to show that fill and fI!1 aTeparallel where 5£, goes through the two points a' = [1,0,I]; b' = [0, I, - I] and 5£2 goes 'through the two points c' = [1,2, -I]; d'= [2, I, I).
2. Find the equation of a line fI!I that goes through the point x' = [I, - 1, 1], intersects another line fI!2, and is perpendicular to fI!2' The line fI! 2 goes through the points a' = 0 and b' = [2, I, 1]-
1. Find the equation of the line in E3 through the point x' = [I, 1,0) and parallel to the line through the two points a' = [0, I, -IJ, b' = [1,0,1]-
24. If xi is a characteristic vector corresponding to a root λi of the matrix A, show that for any positive integer k the vector yi is also a characteristic vector corresponding to the root λi where yi = Akxi.
23. In Prob. 22, find a set of characteristic vectors for A³.
22. In Prob. 12, find A³ and the roots of A³ and demonstrate Theorem 3.2.6.
21. In Prob. 20, show that A and A' do not have the same characteristic vectors.
20. Show that A and A' have the same set of roots, where$$A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix},$$and thus demonstrate Theorem 3.2.5.
19. Consider the two matrices A, Q, where$$A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}; \qquad Q = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}.$$Show that |A|=|Q-1AQ| for these matrices and thus demonstrate Theorem 3.3.1
18. In Prob. 17, verify that P'AP = D, where D is a diagonal matrix with the roots of A as diagonal elements. This is an example of Theorem 3.4.4.
17. Normalize each vector in Prob. 13 and use the two resulting vectors to form the columns of a matrix which we shall denote by P. Verify that P is a 2 x 2 orthogonal matrix.
16. Demonstrate Theorem 3.4.6 by showing that the rank of the matrix A - λ1I in Prob. 12 is one, where λ1 is either of the roots.
15. Verify that the two vectors x and y in Prob. 13 are linearly independent. thus demonstrating Theorem 3.4.5.
14. In Prob. 13, verify that x'y = 0, thus demonstrating Theorem 3.4.3.
13. Find real characteristic vectors x and y associated with each root in Prob. 12.
12. Find the roots of the symmetric matrix A where$$ A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} $$.
11. Show that A in Prob. 10 has no real characteristic vectors.
10. Find the roots of the matrix A where$$\mathbf{A} =\begin{bmatrix}1 & 1 \\-2 & -1\end{bmatrix}.$$
9. In Prob. 8, show that the dimension of S3 is two.
8. Consider the vector space V3 spanned by the four vectors$$\mathbf{v}_1 =\begin{bmatrix}2 \\-1\end{bmatrix};\mathbf{v}_2 =\begin{bmatrix}1 \\0\end{bmatrix};\mathbf{v}_3 =\begin{bmatrix}4 \\-1\end{bmatrix};\mathbf{v}_4 =\begin{bmatrix}-1 \\1\end{bmatrix}.$$and consider the transformation A
7. In Prob. 6, what are conditions on matrices A and B such that, in general, **v = z**In other words, what are conditions such that any n x 1 vector which is trans formed by an n x n matrix A followed by B results in the same vector as when it i transformed by B followed by A?
6. In Prob. 4, suppose x is first transformed to u by **u = Bx** and then u is transformer to v by **v = Au**. Find u and v and demonstrate that **v ≠ z**.
5. In Prob. 4, suppose that the transformation **C = BA** is considered and the vecto x is transformed to w by **w = Cx**. Find w and show that it is equal to z.
4. Consider the 2 x 2 matrices A and B where A = $$\begin{bmatrix} 1 & 1 \\ 2 & -1 \end{bmatrix}$$ ; B = $$\begin{bmatrix} 1 & 1 \\ -2 & 3 \end{bmatrix}$$.If the vector x' = [2, -1] is transformed to y by y = Ax and if y is transformed to z by z = By, find the vectors y and z.
3. For the transformation A in Prob. 1, find the vector y to which the vector x' = [1, -1] is transformed.
2. Find a characteristic vector associated with each root of the matrix A in Prob. 1.
1. Find the roots of the matrix A where A = $$\begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$$.
11. In Prob. 10, find a nonsingular 2 x 2 matrix A that relates the two bases (see Theorem 2.5.6)
10. In Prob. 6, find two different bases for the vector space spanned by the four vectors.
9. In Prob. 6, show that v₂ can be expressed as a linear combination of the o three vectors.
8. Is the vector v in the vector space spanned by the four vectors in Prob. 6, wl v = [1 1 0 1]?
7. In Prob. 6, find a set of two linearly independent vectors.
6. Show that the four vectors below are linearly dependent.$$v_1 = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$$; $$v_2 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$$; $$v_3 = \begin{bmatrix} 0 \\ -6 \\ 2 \\ 0 \end{bmatrix}$$; $$v_4 = \begin{bmatrix} 0 \\ -3 \\ 1 \\ 0 \end{bmatrix}$$.
5. Show that the vector v is in the vector space spanned by v₁ and v₂ where$$v_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$$; $$v_2 = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$$; $$v = \begin{bmatrix} 5 \\ 1 \\ -2 \\ 3 \end{bmatrix}$$.
4. In Prob. 3, generalize to Rn.
3. For any vector x' = [x1, x2, x3] in R3, find scalars c1, c2, c3 such that$$x = \sum_{i=1}^3 c_i e_i$$.
2. In Prob. 1, generalize to Rn.
1. Show that the three vectors$$e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$; $$e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$$; $$e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$are a basis for R3.
If A is defined by A=$$\begin{bmatrix}1 & 1 & 0 & -1 \\2 & 1 & 0 & 1 \\1 & 1 & 0 & 2 \\1 & -1 & 1 & 1\end{bmatrix}$$find det (A) and det (P'AP), where P is defined in Prob. 18, and show that the two are equal. This result demonstrates Theorem 1.8.7.
Find a 3 x 3 matrix A such that det (A) = 1 but such that A is not an orthogonal matrix.

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