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

10. Consider the P-matrix A below.A =$$\begin{bmatrix}3 & -2 \\3 & 4\end{bmatrix}$$.
9. Prove Theorem 11.2.2.
8. If the permutation matrix R is given by$$\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$$, show that R'AR is a P-matrix, where A is given in Prob. 6. This illustrates (4)of Theorem 11.2.2.
7. If D is given below, show that DA, AD, and A + D are P-matrices, where A is given in Prob. 6. This illustrates (i) and (5) of Theorem 11.2.2.D =$$\begin{bmatrix}1 & 0 & 0 \\0 & 3 & 0 \\0 & 0 & 5\end{bmatrix}$$
6. Show that the matrix A below is a P-matrix.A =$$\begin{bmatrix}3 & 10 & -5 \\1 & 4 & -2 \\-1 & 0 & 7\end{bmatrix}$$
5. Find upper and lower triangular matrices R and S, respectively, such that A =RS (where A is defined in Prob. 4) and such that rii > 0, sii > 0. This illustrates Theorem 11.2.1.
4. Show that the matrix A has positive leading principal minors, where A =$$\begin{bmatrix}2 & 4 & -6 \\1 & 8 & 0 \\1 & 4 & 1\end{bmatrix}$$.
3. For the matrix B, find p(B) and note that $p(B) \le p(A)$ since $B = A$, where A is defined in Prob. 1 and B is given below.$$B = \begin{bmatrix}1 & 3 & 1 \\1 & 1 & 3 \\3 & 1 & 1\end{bmatrix}$$This illustrates (4) of Theorem 11.1.1.
2. Find a characteristic vector x of the matrix A in Prob. 1 corresponding to the root $p(A)$ such that x ≥ 0.
1. Find the spectral radius $p(A)$ of the matrix A, where$$A = \begin{bmatrix}1 & 3 & 2 \\2 & 1 & 3 \\3 & 2 & 1\end{bmatrix}$$and illustrate Theorem 11.1.1.
35. Find det (A x B) and det (B x A) if A =$$\begin{bmatrix}2 & 1 \\3 & 4\end{bmatrix}$$, B =$$\begin{bmatrix}1 & 0 & 2 \\2 & 1 & 3 \\2 & 4 & 1\end{bmatrix}$$.
34. In Prob. 33 demonstrate that A x B = (A x I)(I x B).
33. Compute A x B and B x A for the matrices below.A = [1, -1, 0]; B =$$\begin{bmatrix}3 & 1 \\1 & 4 \\2 & 0\end{bmatrix}$$.
32. If A and B are symmetric matrices, show that A x B is symmetric.
31. Use the details of the proof of Theorem 8.6.1 to find a lower triangular matrix R and an upper triangular matrix T such that A = RT where$$\begin{bmatrix}1 & 0 & 0 & 0 & 1 \\0 & 1 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 1 \\0 & 0 & 0 & 1 & 1 \\1 & 1 & 1 & 1 & 1\end{bmatrix}$$.
30. Generalize Prob. 29; that is, find the inverse of the matrix T, where T is a k x k lower triangular matrix, where each element on and below the main diagonal is equal to unity.
29. Find the inverse of the 5 x 5 lower triangular matrix T where$$\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\1 & 1 & 0 & 0 & 0 \\1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 1 & 0 \\1 & 1 & 1 & 1 & 1 \\\end{bmatrix}$$.
26. Find the determinant of the matrix$$A=\begin{bmatrix}0 & a_1 & 0 & 0 \\b_1 & 0 & a_2 & 0 \\0 & b_2 & 0 & a_3 \\0 & 0 & b_3 & 0\end{bmatrix}$$
25. Let the 2k x 2k matrix A be partitioned as follows$$A =\begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}$$where $A_{11}$ is a k x k matrix; further suppose that $A_{21}A_{22}=A_{22}A_{21}$, and let$|A_{22}| eq 0$. Show that det(A) = det($A_{11}A_{22}-A_{12}A_{21}$).(Use Theorem
24. Evaluate the determinant of A where $x_i = i$ and $k = 4$.$$A =\begin{bmatrix}1 & 1 & 1 & ... & 1 \\x_1 & x_2 & x_3 & ... & x_k \\x_1^2 & x_2^2 & x_3^2 & ... & x_k^2 \\. & . & . & ... & . \\x_1^{k-1} & x_2^{k-1} & x_3^{k-1} & ... & x_k^{k-1}\end{bmatrix}$$
23. Use Theorem 8.2.1, to find the determinant of the matrix A where A =$$\begin{bmatrix}1 & 3 & 1 & 3 \\4 & 2 & 2 & 1 \\4 & 2 & 2 & 3 \\3 & 1 & 4 & 1\end{bmatrix}$$
22. Evaluate the determinant of the matrix B where B is defined by?
21. Use Theorem 8.4.3 to evaluate the determinant of the matrix V in Example 8.3.3
20. Use Theorem 8.9.3 to find the characteristic roots of the matrix in Prob. 19.
19. Use Theorem 8.9.3 to find the determinant of B where$$\begin{aligned}B &= \begin{bmatrix}2 & 2 & 3 \\2 & 5 & 6 \\3 & 6 & 10\end{bmatrix}\end{aligned}$$Note that B = I + bb' where b' = [1, 2, 3].
18. Prove Theorem 8.9.5 by using Theorem 8.2.1.
17. Use Theorem 8.2.1 to find the inverse of A where$$\begin{aligned}A &= \begin{bmatrix}1 & 0 & 0 & 0 & 3 \\0 & 1 & 0 & 0 & 2 \\0 & 0 & 1 & 0 & 1 \\0 & 0 & 0 & 1 & 2 \\3 & 2 & 1 & 2 & 4\end{bmatrix}\end{aligned}$$
16. Find A-1 in Prob. 14.
15. Find the inverse of the matrix in Prob. 3.
14. Find the determinant and characteristic roots of the matrix A where$$\begin{aligned}A &= \begin{bmatrix}0 & I & I & \dots & I \\I & 0 & I & \dots & I \\I & I & 0 & \dots & I \\\vdots & \vdots & \vdots & & \vdots \\I & I & I & \dots & 0\end{bmatrix}\end{aligned}$$each identity has size k x k,
13. Extend Prob. 12 to the case in which there are n² block matrices and the order of each is k x k.
12. Find the inverse of the triangular matrix T where$$\begin{aligned}T &= \begin{bmatrix} I & J & J \\ 0 & I & J \\ 0 & 0 & I \end{bmatrix}\end{aligned}$$and each submatrix is of order k x k.
11. If$$\begin{aligned}B &= \begin{bmatrix} 6 & 6 & 6 & 6 & 6 \\ 6 & 8 & 8 & 8 & 8 \\ 6 & 8 & 3 & 3 & 3 \\ 6 & 8 & 3 & 2 & 2 \\ 6 & 8 & 3 & 2 & 4 \end{bmatrix},\end{aligned}$$find B⁻¹.
11. If$$B = \begin{bmatrix}6 & 6 & 6 & 6 & 6 \\6 & 8 & 8 & 8 & 8 \\6 & 8 & 3 & 3 & 3 \\6 & 8 & 3 & 2 & 2 \\6 & 8 & 3 & 2 & 4\end{bmatrix}$$
10. Generalize Probs. 8 and 9 to a $k \times k$ matrix.
9. If the conditions on the $a_i$ in the matrix A in Prob. 8 are such that A is nonsingular, find A⁻¹.
8. Let$$A = \begin{bmatrix}a_1 & a_1 & a_1 & a_1 \\a_1 & a_2 & a_2 & a_2 \\a_1 & a_2 & a_3 & a_3 \\a_1 & a_2 & a_3 & a_4\end{bmatrix};$$what are the conditions on the $a_i$ so that A is nonsingular?
7. Find the determinant of the matrix.$$C = \begin{bmatrix}al & J & J & J \\J & al & J & J \\J & J & al & J \\J & J & J & al\end{bmatrix}$$where $a eq 0$ and each matrix has dimension $n \times n$. What are the conditions on$a$ and $n$ to insure that the inverse exists?
6. Find the inverse of the matrix C where$$\begin{bmatrix} a_1 & a_2 & a_2 & a_2\\ a_2 & a_3 & a_4 & a_4\\ a_2 & a_4 & a_3 & a_4\\ a_2 & a_4 & a_4 & a_3 \end{bmatrix}$$if thea, are such that the inverse exists.
5. In Theorem 8.2.1, suppose n₁ = n₂ and B₁₁ = B₂₂ = 0. State a result for the existence of B⁻¹.
4. Let a k x k matrix C be defined by Eq. (8.3.13).(a) Find the conditions on the constantsa, b, and k such that C is positive definite.(b) Find the conditions on the constantsa, b, and k such that C is positive semi-definite.(c) Find the conditions on the constantsa, b, and k such that C² = C.
3. If B =$$\begin{bmatrix} -I & A \\ A & -I \end{bmatrix}$$, where A is an m x m symmetric matrix such that A² = A, show that |B| = (-1)ᵐ.
2. In Prob. 1, if ad - b² ≠ 0, find B⁻¹.
1. If B =$$\begin{bmatrix} aI & bI \\ bI & dI \end{bmatrix}$$, where each identity matrix is of size m x m, find the characteristic roots of B.
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 $B_{11}$ is an $n \times n$ submatrix. Show that(1) $b=0$,(2) $b_{12}=(1/n)1$,(3) $1'B_{11}=0$,(4) $AB_{11}$ and $B_{11}A$ 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 $A^k + J$ is nonsingular where A is defined in Prob. 38 and $k$ is any positive integer.
38. Let A be a symmetric $n \times n$ 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 $m \times n$ matrix and if the rank of A is $n$, show that $A^+$ is unique.
36. Consider the linear model y = Xẞ + e and the normal equations?
35. Prove Theorem 7.6.6.
34. Let A be any matrix and let (AtA)t be any L-inverse of AtA. Show that A-1 = A(AtA)t.
33. If A is a symmetric matrix show that A-1 = A(At)2 for any L-inverse of A.
32. In Prob. 31 show that B = AA.
31. Prove that B is symmetric idempotent where B = A(AtA)tAt and where (AtA)tis any c-inverse of AtA.
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$$B = \frac{1}{2}[A^t + (A^t)']$$and where At is any c-inverse of A.
28. If A is symmetric show that for any c-inverse At of A the matrix (At)' is also a c-inverse of A.
27. If A is a symmetric matrix and At is any c-inverse of A, show that At 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 m x n matrix and define B by B = (AtA)tAt where (AtA)t is any c-inverse of AtA. Show that ABA = A; BAB = B; AB is symmetric.
24. Prove that if A is nonsingular, then A-1 = At = At = A-1.
23. Prove that if AAt = I then AAt = I for all L-inverses of A.
22. Prove that for any matrix A if there is an L-inverse At such that AAt = I then AAt = I.
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'XB = 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.
18. Show that the system of equations below is consistent and that a unique solution does not exist.$$\begin{aligned}4x_1 + 2x_2 + 2x_3 &= 3, \\2x_1 + 2x_2 &= 0,\end{aligned}$$
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 $x_1$ and $x_2$ be any solutions to $Ax = g$. Show that the vector $x_1 - x_2$ is orthog-onal to the rows of A.
11. Let $x_0$ be a solution to $Ax = g$ where $x_0 = A^T g$. Show that y is a solution where$y=x_0 + z$ for all vectors z that belong to the orthogonal complement of the column space of $A^T$.
10. Prove that for any matrix A if $AA^T=I$, then $AA^T=I$ for each c-inverse of A.
8. If x1, x2, ..., xr are solutions to the system Ax = g, show that y = ∑i=1rcixi
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 - AAc)hi.
6. Find any two distinct solutions x1 and x2 to the system in Prob. I and demonstrate that 3x1 + 3x2 is also a solution.
5. In Prob. 1 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-1 =H-1H.
75. Let A be an m x n matrix and B be an n x m matrix. Show that if ABB-1 =A and B-1B = AB, then A = B-1.
74. If A, B, and X are m x n, k x n, and m x k matrices, respectively, show that XX-1 = AA-1 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)-1 = A-1B-1A'-1.
72. Let A be an m x n matrix. Show that A-1 = A'(AA')A(A'A)A', where (A'A)-1and (AA')-1 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)-1 = B-1A-1 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 =AA'A' + (I - AA')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)' =B'A' (where (AB)', A', B' are any c-inverses of the respective matrices) if and only if A'ABB' is idempotent.
65. Let A be an n x n symmetric matrix such that A² = mA. Show that B =1 mA is a g-inverse of 1/mA.
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.

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