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Matrices With Applications In Statistics 2nd Edition Franklin A Graybill - Solutions
17. Find the g-inverse of the matrix A where--- OCR End ---
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.
29. In Prob. 28, if and P is nonsingular, show that P-1A' is a c-inverse of AP.
30. In Prob. 28, if PP' = I, show that P'A' is a c-inverse of AP.
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'.
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'.
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'.
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.
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?
18. Find the g-inverse of the matrix A where$$A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & -2 & 0 \\ 3 & 3 & -3 \end{bmatrix}$$
19. Let A be an mxn given matrix, and let X be any nx 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 A = YAX.
20. Find a solution to the system of equations 2x1-x2+x3=8, x1+2x2-x3=-5.
21. If A is a given mxn matrix, find conditions on a matrix X so that the system AX= I is consistent.
22. Prove Theorem 6.3.3.
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.
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'.
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).
36. Find a c-inverse of the matrix A in Prob. 6.
20. Prove Theorem 5.6.4.
12. In Prob. II, find the point on 11 where 2 intersects {?
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.
14. In Prob. 13, show that this value of d is the minimum distance between 0 and the plane fi'.
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].
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
17. In Prob. 16, sketch the three lines.
18. In Prob. 16, find the angle between t and fil.
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]-
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.
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]-
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).
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
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].
6. Find the direction angles of II in Prob, 5.
7. Find the direction angles of the line through the points a' = [2,0,-I]and b' = [I, -1,3],
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].
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.
20. Consider the plane &'1 through the points ai' 32, 0 wherea, = [I, 1,0,1], 32 = [I, -I, 1,0).Find the projection ?
1. If a₁, a₂, a₃ span S₁ and b₁, b₂, b₃ span S₂, find a basis for S₁ ∩ S₂.$$a_1 = \begin{bmatrix}1 \\ 3 \\ -1 \\ 2\end{bmatrix}; a_2 = \begin{bmatrix}0 \\ -1 \\ 2 \\ 1 \end{bmatrix}; a_3 = \begin{bmatrix}2 \\ 7 \\ 2 \\ -3 \end{bmatrix};$$$$b_1 = \begin{bmatrix}1 \\ 8 \\ 0 \\
12. Find the dimension of the column space of A where$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3 \\1 & 2 & 0\end{bmatrix}$$
13. If A and B are nonsingular n x n matrices, prove that they have the same column space.
14. Let A and B be m x n matrices of rank n where n < m. Show that A and B may not have the same column space.
15. Prove Theorem 5.4.5.
16. Let A and B be defined by$$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.
17. In Prob. 16, find a matrix C such that AC = B.
18. Prove Corollary 5.3.3.
19. Prove Theorem 5.6.2.
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 x, and x2 where x\ = [1,1,0, -1]; x2 = (0, I, 1, IJ; the vector y is defined by y' = (I, 2, - I, I]. Find vectors ZI and Zl such that y =ZI + Z2 where Zt is in Sand Z2 is in SJ..
2. In Prob. 1, find a basis for S₁ ⊕ S₂.
3. If {a₁, ..., aₚ} spans S₁ and {b₁, ..., bₛ} spans S₂, prove that {a₁, ..., aₚ, b₁, ..., bₛ} spans S₁ ⊕ S₂.
4. In Prob. 1, find the dimension of S₁ ∩ S₂.
5. In Prob. 1, find the dimension of S₁ ⊕ S₂.
6. Use the results of Probs. 4 and 5 to demonstrate Theorem 5.2.5.
7. Prove Theorem 5.2.5.
8. Find a basis for the orthogonal complement of the subspace in E₃ spanned by x where x' = [1, 1, -1].
9. Find the vector z and the scalar λ such that y = λx + z--- OCR End ---
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]-
37. Show rank (A') ≥ rank (A) for any c-inverse of A. (See Theorem 6.6.8.)
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.
14. In Prob. 13 find the BAS.
15. In Prob. 13 find an LSS.
16. Show that the system below is inconsistent.$$\begin{aligned}x_1 + x_2 + x_4 &= 1,\\x_1 + x_2 + x_5 &= 2,\\x_1 + x_3 + x_4 &= 1,\\x_1 + x_3 + x_5 &= 3.\end{aligned}$$
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.
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.
19. In Prob. 18 find a 2 x 3 matrix G such that Gx is unique for any solution vector x.
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.
4. In Prob. I find the number of linearly independent solution vectors.
5. In Prob. I find a linearly independent set of solution vectors.
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.
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.
8. If x1, x2,..., xr are solutions to the system Ax = g, show that y = ∑i=1rcixi--- OCR End ---
9. Prove that for any matrix A if there is a c-inverse AC, such that AAc = I, then AA-=1.
10. Prove that for any matrix A if AA⁻¹ = I, then AA⁻¹ = I for each c-inverse 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'.
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'.
21. Find an L-inverse of the matrix A where A =$$\begin{bmatrix}1 & 2 & 7 \\1 & 1 & 5 \\-1 & 2 & 1 \\2 & 1 & 8\end{bmatrix}$$.
32. In Prob. 31 show that B = AA.
33. If A is a symmetric matrix show that A
34. Let A be any matrix and let (A'A)A'A(A'A).
35. Prove Theorem 7.6.6.=A(A) for any L-inverse of A.be any L-inverse of A'A. Show that
36. Consider the linear model y Xẞe and the normal equations
37. If A is an *m x n* matrix and if the rank of A is *n*, show that A-1 is unique.
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.
39. Show that Ak + J is nonsingular where A is defined in Prob. 38 and *k* is any positive integer.
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.
22. Prove that for any matrix A if there is an L-inverse A¹ such that AA² = I then AAI.
23. Prove that if AA I then AAI for all L-inverses of A.
24. Prove that if A is nonsingular, then A A A
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.
26. In Prob. 25 show that B may not always be a g-inverse of A, since BA may not always be symmetric.
27. If A is a symmetric matrix and A is any c-inverse of A, show that A is not necessarily symmetric.
28. If A is symmetric show that for any c-inverse A of A the matrix (A')' is also a c-inverse of A.
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.
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