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Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

Let A be the adjacency matrix of a digraph D. a. If row I of A2 is all zeros, what does this imply about D? b. If column j of A2 is all zeros, what does this imply about D?
Figure is the digraph of tournament with six players, P1 to P6. Using adjacency matrices rank the player first by determining wins only and then by using the notion of combined wins and indirect wins.
Figure is a digraph representing a food web in a small ecosystem. A directed edge from a to b indicates that a has b as a source of food. Construct the adjacency matrix A for this digraph and use it to answer the following questions.a. Which species has the most direct sources of food? How does A
Five people are all connected by e-mail. Whenever one of them hears a juicy piece of gossip, he or she passes it along by e-mailing it to someone else in the group according to Table. a. Draw the digraph that models this gossip network and find its adjacency matrix A. Sender Recipients Ann
Let A be the adjacency matrix of a graph G a. By induction, prove that for all n ≥ 1, the (i, j) entry of A" is equal to the number of n-paths between vertices i and j. b. How do the statement and proof in part (a) have to be modified if G is a digraph?
A graph is called bipartite if its vertices can be subdivided into two sets U and V such that every edge has one endpoint in U and the other endpoint in V. For example the graph in Exercise 48 is bipartite with U = {v1, v2, v3} and V = {v4, v5}. In Exercises determine whether a graph with given
a. Prove that a graph is bipartite if and only if its vertices can be labeled so that its adjacency matrix can be partitioned asb. Using the result in part (a), prove that a bipartite graph has no circuits of odd length.
Suppose that the weather in a particular region behaves according to a Markov chain. Specifically, suppose that the probability that tomorrow will be a wet day is 0.662 if today is wet and 0.250 if today is dry. The probability that tomorrow will be a dry day is 0.750 if today is dry and 0.338 if
If possible, express the matrixAs a product of elementary matrices
If A is a square matrix such that A3 = O, show that (I - A)-1 = I + A + A2
Find an LU factorization of
Find bases for the row space, column space, and null space of
Suppose matrices A and B are row equivalent. Do they have the same row space? Why or why not? Do A and B have the same column space? Why or why not?
If A is an invertible matrix, explain why A and AT must have same null space. Is this true if A is a noninvertible square matrix? Explain
Let A be an m × n matrix with linearly independent columns. Explain why ATA must be an invertible matrix. Must AAT also be invertible? Explain.
Find a linear transformation T : R2 †’ R2 such thatAnd
Find the standard matrix of the linear transformation T : R2 → R2 that corresponds to a counterclockwise rotation of 45o about the origin followed by a projection onto the line y = -2x
Suppose that T : Rn → Rn is a linear transformation and suppose that v is a vector such that T(v) ≠ 0 but T2 = T o T). Prove that v and T(v) are linearly independent.
If A is a matrix such thatFind A.
In Exercises 1-6, show that v is an eigenvector of A and find the corresponding eigenvalue.1.2. 3. 4. 5.
In Exercises 1 - 3, find the eigenvalues and eigenvectors of A geometrically.1.2. 3.
In Exercises 1-2, the unit vectors x in R2 and their images Ax under the action of a 2 X 2 matrix A are drawn head-to-tail, as in Figure 4.7. Estimate the eigenvectors and eigenvalues of A from each ''eigenpicture''.1.2.
In Exercises 1-3, show that A is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue.1.2. 3.
In Exercises 1-2, use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. Illustrate the eigenspaces and the effect of multiplying eigenvectors by A as in Figure 4. 8.1.2.
In Exercises 1-2, find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces.1.2.
In Exercises 1-4, find all of the eigenvalues of the matrix A over the indicated Zp.1.2. 3. 4.
(a) Show that the eigenvalues of the 2 Ã— 2 matrix
Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) Two distinct real eigenvalues, (b) One real eigenvalue, and (c) No real eigenvalues.
Show that the eigenvalues of the upper triangular Matrixare Î» = a and Î» = d, and find the corresponding
Let a and b be real numbers. Find the eigenvalues and corresponding eigenspaces ofover the complex numbers.
Compute the determinants in Exercises 1-5 using cofactor expansion along the first row and along the first column.1.2. 3. 4. 5.
In Exercises 1-3, compute the indicated 3 Ã— 3 determinants using the method of Example 4. 9.1. The determinant in Exercise 62. The determinant in Exercise 8 3. The determinant in Exercise 11
Verify that the method indicated in (2) agrees with Equation (1) for a 3 × 3 determinant.
Verify that definition (4) agrees with the definition of a 2 × 2 determinant when n = 2.
Prove Theorem 4.2. [A proof by induction would be appropriate here.] In theorem 4.2 The determinant of a triangular matrix is the product of the entries on its main diagonal. Specifically, if A = [aij) is an n × n triangular matrix, then det A = a11 a22 ··· ann
In Exercises 1-4, evaluate the given determinant using elementary row and/or column operations and Theorem 4.3 to reduce the matrix to row echelon form.1. The determinant in Exercise 12. The determinant in Exercise 9 3. The determinant in Exercise 13 4. The determinant in Exercise 14
In Exercises 1-5, use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.1.2. 3. 4. 5.
Find the determinants in Exercises 1-5, assuming that1. 2. 3. 4. 5.
In Exercises 45 and 46, use Theorem 4. 6 to find all values of k for which A is invertible1.2.
In Exercises 1-5, assume that A and B are n × n matrices with det A = 3 and det B = - 2. Find the indicated determinants. 1. det (AB) 2. det(A2) 3. det(B-1A) 4. det(2A) 5. det(3BT) 6. det(AAT)
In Exercises 1-3, A and B are n X n matrices. 1. Prove that det(AB) = det(BA). 2. If B is invertible, prove that det(B-1AB) = det(A). 3. If A is idempotent (that is, A2 = A), find all possible values of det(A).
In Exercises 1-4, use Cramer's Rule to solve the given linear system. 1. x + y = 1 x - y = 2 2. 2x - y = 5 x + 3y = - 1 3. 2x + y + 3z = 1 y + z = 1 z = 1 4. x + y - z = 1 x + y + z = 2 x - y = 3
In Exercises 1-4, use Theorem 4.12 to compute the inverse of the coefficient matrix for the given exercise. 1. Exercise 57 x + y = 1 x - y = 2 2. Exercise 58 2x - y = 5 x + 3y = - 1 3. Exercise 59 2x + y + 3z = 1 y + z = 1 z = 1 4. Exercise 60 x + y - z = 1 x + y + z = 2 x - y = 3
If A is an invertible n Ã— n matrix, show that adj A is also invertible and that
Verify that if r < s, then rows r and s of a matrix can be interchanged by performing 2(s - r) - 1 interchanges of adjacent rows.
Prove that the Laplace Expansion Theorem holds for column expansion along the jth column.
Let A be a square matrix that can be partitioned aswhere P and S are square matrices. Such a matrix is said to be in block (upper) triangular form. Prove that det A = (det P)(det S)
Compute the determinants in Exercises 1-5 using cofactor expansion along any row or column that seems convenient.1.2. 3. 4. 5.
(a) Give an example to show that if A can be partitioned aswhere P, Q, R, and S are all square, then it is not necessarily true that det A = (det P) (det S) - (det Q) (det R) (b) Assume that A is partitioned as in part (a) and that P is invertible. Let Compute det (BA) using Exercise 69 and use the
In Exercises 1 - 2, compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspaces of A,(d) The algebraic and geometric multiplicity of each eigenvalue.1.2.
In Exercises 1 and 2, A is a 2 Ã— 2 matrix with eigenvectorsAnd corresponding to eigenvalues Î»1 = t and Î»2 = 2, respectively, and 1. Find A10x. 2. Find Akx. What happens as k becomes large (i.e., k ( ˆž)?
In Exercises 1 and 2, A is a 3 Ã— 3 matrix with eigenvectorscorresponding to eigenvalues Î»1 = - 1/3, Î»2 = 1/3, and Î»3 = 1 , respectively, and 1. Find A20x. 2. Find Akx. What happens as k becomes large (i.e., k ( ˆž)?
(a) Show that, for any square matrix A, AT and A have the same characteristic polynomial and hence the same eigenvalues. (b) Give an example of a 2 × 2 matrix A for which AT and A have different eigenspaces.
Let A be a nilpotent matrix (that is, Am = 0 for some m > 1). Show that λ = 0 is the only eigenvalue of A.
Let A be an idempotent matrix (that is, A2 = A). Show that λ = O and λ = 1 are the only possible eigenvalues of A.
If v is an eigenvector of A with corresponding eigenvalue λ and c is a scalar, show that v is an eigenvector of A - cI with corresponding eigenvalue λ - c.
(a) Find the eigenvalues and eigenspaces of(b) Using Theorem 4. 18 and Exercise 22, find the eigenvalues and eigenspaces of A-1, A - 2I, and A + 2I.
Let A and B be n × n matrices with eigenvalues λ and μ, respectively. (a) Give an example to show that λ + μ need not be an eigenvalue of A + B. (b) Give an example to show that λμ need not be an eigenvalue of AB. (c) Suppose λ and μ correspond to the same eigenvector x. Show that, in this
If A and B are two row equivalent matrices, do they necessarily have the same eigenvalues? Either prove that they do or give a counterexample.Let p(x) be the polynomialThe companion matrix of p(x) is the n Ã— n matrix
Find the companion matrix of p(x) = x2 - 7x + 12 and then find the characteristic polynomial of C(p).
Find the companion matrix of p(x) = x3 + 3x2 - 4x + 12 and then find the characteristic polynomial of C(p).
(a) Show that the companion matrix C(p) of p(x) = x2 + ax + b has characteristic polynomial Î»2 + aÎ» + b.(b) Show that if A is an eigenvalue of the companion matrix C(p) in part (a), thenis an eigenvector of C(p) corresponding to Î».
(a) Show that the companion matrix C(p) of p (x) = x3 + ax2 + bx + c has characteristic polynomial - (λ3 + aλ2 + bλ + c). (b) Show that if A is an eigenvalue of the companion
Construct a nontriangular 2 × 2 matrix with eigenvalues 2 and 5.
Construct a nontriangular 3 × 3 matrix with eigenvalues - 2, 1, and 3.
(a) Use mathematical induction to prove that, for n ‰¥ 2, the companion matrix C(p) of p (x) = xn + an-1xn-1 + ··· + a1x + a0 has characteristic polynomial (-1) np (A).(b) Show that if A is an eigenvalue of the companion matrix C(p) in Equation (4), then an eigenvector
Verify the Cayley-Hamilton Theorem forThat is, find the characteristic polynomial cA(Î») of A and show that cA(A) = 0. An important theorem in advanced linear algebra says that if cA (Î») is the characteristic polynomial of the matrix A, then cA (A) = 0 (in words, every
Verify the Cayley-Hamilton Theorem forThe Cayley-Hamilton Theorem can be used to calculate powers and inverses of matrices. For example, if A is a 2 x 2 matrix with characteristic polynomial cA(Î») = Î»2 + aÎ» + b, then A2 + aA + bl = 0, so A2 = - aA - bl and
For the matrix A in Exercise 33, use the CayleyHamilton Theorem to compute A2, A3, and A4 by expressing each as a linear combination of I and A.In exercise 33An important theorem in advanced linear algebra says that if cA (Î») is the characteristic polynomial of the matrix A, then cA
For the matrix A in Exercise 34, use the CayleyHamilton Theorem to compute A3 and A4 by expressing each as a linear combination of I, A, and A2.
For the matrix A in Exercise 33, use the CayleyHamilton Theorem to compute A-1 and A-2 by expressing each as a linear combination of I and A.In exercise 33An important theorem in advanced linear algebra says that if cA (Î») is the characteristic polynomial of the matrix A, then cA (A) =
For the matrix A in Exercise 34, use the Cayley-Hamilton Theorem to compute A - 1 and A -2 by expressing each as a linear combination of I, A, and A2.
Show that if the square matrix A can be partitioned aswhere P and S are square matrices, then the characteristic polynomial of A is cA(Î») = cp(Î») cS(Î»).
Let λ1, λ2, ..., λn be a complete set of eigenvalues (repetitions included) of the n × n matrix A. Prove that det(A) = λ1 λ2 ··· An and tr(A) = λ1 + λ2 + ··· + λn
Let A and B be n × n matrices. Prove that the sum of all the eigenvalues of A + B is the sum of all the eigenvalues of A and B individually. Prove that the product of all the eigenvalues of AB is the product of all the eigenvalues of A and B individually.
In Exercises 1-4, show that A and B are not similar matrices1.2. 3. 4.
In Exercises 1-3, use the method of Example 4.29 to compute the indicated power of the matrix.1.2. 3.
In Exercises 24-29, find all (real) values of k for which A is diagonalizable.1.2. 3.
In general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. In Exercises 1-2 show that A and are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1
Prove that if A is diagonalizable, so is AT.
Let A be an invertible matrix. Prove that if A is diagonalizable, so is A-1.
Prove that if A is a diagonalizable matrix with only one eigenvalue λ, then a is of the form A = λl. (Such a matrix is called a scalar matrix).
Let A and B be n × n matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB = BA.
Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and B are the same.
Prove that if A is a diagonalizable matrix such that every eigenvalue of A is either 0 or 1, then A idempotent (that is, A2 = A).
In Exercises 5-7, a diagonalization of the matrix A is given in the form P-1 AP = D. List the eigenvalues of A and bases for the corresponding eigenspaces.1.2. 3.
Let A be a nilpotent matrix (-at is, Am = O for some m > 1). Prove -at if A is diagonalizable, -en A must be -e zero matrix.
Suppose that A is a 6 × 6 matrix with characteristic polynomial cA(λ) = (1 + λ) (1 - λ)2 (2 - λ)3. a. Prove that it is not possible to find three linearly independent vectors v1, v2, v3 in R6 such that Av1 = v1, Av2 = v2, and Av3 = v3. b. If A is diagonalizable, what are the dimensions of the
In Exercises 8-15, determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D1.2. 3.
In Exercises 1-3, a matrix A is given along with an iterate x5, produced as in Example 4.30.(a) Use these data to approximate a dominant eigenvector whose first component is 1 and a corresponding dominant eigenvalue. (Use three-decimal-place accuracy.)(b) Compare your approximate eigenvalue in part
In Exercises 1 and 2, use the power method to approximate the dominant eigenvalue and eigenvector of A to two-decimal-place accuracy. Choose any initial vector you like (but keep the first Remark after Example 4.31 in mind!) and apply the method until the digit in the second decimal place of the
In Exercises 1-4, to see how the Rayleigh quotient method approximates the dominant eigenvalue more rapidly than the ordinary power method, compute the successive Rayleigh quotients R(xi) for i = 1, ... , k for the matrix A in the given exercise.1. Exercise 112. Exercise 12 3. Exercise 13 4.
The matrices in Exercises 1-3 either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector x0, performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening.1.2. 3.
In Exercises 1-3, the power method does not converge to the dominant eigenvalue and eigenvector. Verify this, using the given initial vector x0. Compute the exact eigenvalues and eigenvectors and explain what is happening.1.2. 3.
In Exercises 1-4, apply the shifted power method to approximate the second eigenvalue of the matrix A in the given exercise. Use the given initial vector x0, k iterations, and three-decimal-place accuracy.1. Exercise 92. Exercise 10 3. Exercise 13 4. Exercise 14
In Exercises 1-2, apply the inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue that is smallest in magnitude. Use the given initial vector x0, k iterations, and three-decimal-place accuracy.1. Exercise 92. Exercise 10 3. Exercise 7, 4. Exercise 14
In Exercises 1-4, use the shifted inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue closest to a. 1. Exercise 9, α = 0 2. Exercise 12, α = 0 3. Exercise 7, α = 5 4. Exercise 13, α = - 2
Exercise 32 in Section 4.3 demonstrates that every polynomial is (plus or minus) the characteristic polynomial of its own companion matrix. Therefore, the roots of a polynomial p are the eigenvalues of C (p). Hence, we can use the methods of this section to approximate the roots of any polynomial
Let λ be an eigenvalue of A with corresponding eigenvector x. If a ≠ λ and α is not an eigenvalue of A, show that 1 / (λ - α) is an eigenvalue of (A - al)-1 with corresponding eigenvector x. (Why must A = αI be invertible??
If A has a dominant eigenvalue λ1, prove that the eigenspaces Eλ1 is one-dimensional.

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