New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
linear algebra

Linear Algebra with Applications 7th edition Steven J. Leon - Solutions

Letand Is it possible to perform the block multiplications of AAT and ATA? Explain.
Let A be an m × n matrix, V an n × r matrix, and B an m × r matrix. Show that AX = B if and only if Axj = bj, j = 1,..., r
Let A be an n × n matrix and let D be a n × n diagonal matrix. (a) Show that D = (d11e1, d22e2,..., dnnen). (b) Show that AD = (d11a1, d22a2, dnnan).
If the row echelon form of A involves free variables, then the system Ax = b will have infinitely many solutions.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an
Let A be a 4 Ã— 3 matrix with a2 = a3. If b = a1 + a2 + a3, then the system Ax = b will have infinitely many solutions.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false
An n Ã— n matrix A is nonsingular if and only if the reduced row echelon form of A is I (the identity matrix).In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement,
If A and B are nonsingular n Ã— n matrices, then A + B is also nonsingular and (A + B)-1 = A-1 + B-1.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an
If A and B are nonsingular n Ã— n matrices, then AB is also nonsingular and (AB)-1 = A-1B-1.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to
If A and B are n Ã— n matrices, then (A - B)2 = A2 - 2AB + B2.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not
If AB = AC and A ‰ O (the zero matrix), then B = C.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
The product of two elementary matrices is an elementary matrix.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always
If x and y are nonzero vectors in Rn and A = xyT, then the row echelon form of A will have exactly one nonzero row.In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an
Find all solutions to the linear system x1 - x2 + 3x3 + 2x4 = 1 -x1 + x2 - 2x3 + x4 = -2 2x1 - 2x2 + 7x3 + 7x4 = 1
Let E and F be n × n elementary matrices and let C = EF. Is C nonsingular? Explain.
Let A and B be 10 Ã— 10 matrices that are partitioned into submatrices as follows:(a) If A11 is a 6 Ã— 5 matrix, and B11 is a k Ã— r matrix, what conditions, if any, must k and r satisfy in order to make the block multiplication of A times B possible? (b)
(a) A linear equation in 2 unknowns corresponds to a line in the plane. Give a similar geometric interpretation of a linear equation in 3 unknowns. (b) Given a linear system consisting of 2 equations in 3 unknowns, what are the possible number of solutions. Give a geometric explanation of your
Let Ax = b be a system of n linear equations in n unknowns and suppose that x1 and x2 are both solutions and x1 ≠ x2. (a) How many solutions will the system have? Explain. (b) Is the matrix A nonsingular? Explain.
Let A be a matrix of the formwhere Î± and Î² are fixed scalars not both equal to 0. (a) Explain why the system must be inconsistent. (b) How can one choose a nonzero vector b so that the system Ax = b will be consistent? Explain.
Given(a) Find an elementary matrix E such that EA = B. (b) Find an elementary matrix F such that AF = C.
Let A be a 3 × 3 matrix and let b = 3a1 + a2 + 4a3 Will the system Ax = b be consistent? Explain.
Let A be a 3 × 3 matrix and suppose that a1 - 3a2 + 2a3 = 0 (the zero vector) Is A nonsingular? Explain.
Given the vectorIs it possible to find 2 Ã— 2 matrices A and B so that A ‰ B and Ax0 = Bx0? Explain.
Let A and B be symmetric n × n matrices and let C = AB. Is C symmetric? Explain.
Use MATLAB to generate random 4 × 4 matrices A and B. For each of the following compute A1, A2, A3, A4 as indicated and determine which of the matrices are equal. You can use MATLAB to test whether two matrices are equal by computing their difference. (a) A1 = A * B, A2 = B * A, A3 = (A′ *
Set B = [-1, -1; 1, 1] and A = [zeros(2), eye(2); eye(2), B] and verify that B2 = O. (a) Use MATLAB to compute A2, A4, A6, and A8. Make a conjecture as to what the block form of A2k will be in terms of the submatrices I, O, and B. Use mathematical induction to prove that your conjecture is true for
(a) The MATLAB commandsA = floor(10 * rand(6)), B = A€² * Awill result in a symmetric matrix with integer entries. Why? Explain. Compute B in this way and verify these claims. Next, partition B into four 3 Ã— 3 submatrices. To determine the submatrices in MATLAB, setB11 = B(l
Set A = floor(10 * rand(6)). By construction, the matrix A will have integer entries. Let us change the sixth column of A so as to make the matrix singular. Set B = A', A(:,6) = -sum(B(l : 5, :))' (a) Set x = ones(6, 1) and use MATLAB to compute Ax. Why do we know that A must be singular? Explain.
Construct a matrix as follows. Set B = eye(10) - triu(ones(10), 1) Why do we know that B must be nonsingular? Set C = inv(B) and x = C(:, 10) Now change B slightly by setting B(10, 1) = -1/256. Use MATLAB to compute the product Bx. From the result of this computation, what can you conclude about
Generate a matrix A by setting A = floor(10 * rand(6)) and generate a vector b by setting b = floor(20 * rand(6, 1)) - 10 (a) Since A was generated randomly, we would expect it to be nonsingular. The system Ax = b should have a unique solution. Find the solution using the "\" operation. Use MATLAB
Consider the graph (a) Compute A4, A6, A8 and answer the questions in part (b) for walks of lengths 4, 6, and 8. Make a conjecture as to when there will be no walks of even length from vertex Vi to vertex Vj. (b) Compute A3, A5, A7 and answer the questions from part (b) for walks of lengths 3, 5,
The following table describes a seven-stage model for the life cycle of the loggerhead turtle.The corresponding Leslie matrix is given by Suppose that the number of turtles in each stage of the initial turtle population is described by the vector x0 = (200,000 130,000 100,000 70,000 500 400
Use mathematical induction to prove that if A is an (n + 1) × (n + 1) matrix with two identical rows then det(A) = 0.
Let A and B be 2 × 2 matrices. (a) Does det(A + B) = det(A) + det(B)? (b) Does det(AB) = det(A) det(B)? (c) Does det(AB) = det(BA)? Justify your answers.
Let A and B be 2 Ã— 2 matrices and let(a) Show that det(A + B) = det(A) + det(B) + det(C) + det(D). (b) Show that if B = EA then det(A + B) = det(A) + det(B).
Let A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 whenever |i - j | > 1). Let B be the matrix formed from A by deleting the first two rows and columns. Show that det(A) = a11 det(M11) - a212det(B)
Let A be a 3 × 3 matrix with a11 = 0 and a21 ≠ 0. Show that A is row equivalent to I if and only if -a12a21a33 + a12a31a23 + a13a21a32 - a13a31a22 ≠ 0
Write out the details of the proof of Theorem 2.1.3.
Prove that if a row or a column of an n × n matrix A consists entirely of zeros then det(A) = 0.
Let A and B be row equivalent matrices, and suppose that B can be obtained from A using only row operations I and III. How do the values of det(A) and det(B) compare? How will the values compare if B can be obtained from A using only row operation III? Explain your answers.
Let A be an n × n matrix. Is it possible for A2 + I = O in the case where n is odd? Answer the same question in the case where n is even.
Consider the 3 Ã— 3 Vandermonde matrix(a) Show that det(V) = (x2 - x1)(x3 - x1)(x3 - x2). (b) What conditions must the scalars x1, x2, x3 satisfy in order for V to be nonsingular?
Let A and B be n × n matrices. Prove that the product AB is nonsingular if and only if A and B are both nonsingular.
Let A and B be n × n matrices. Prove that if AB = I then BA = I. What is the significance of this result in terms of the definition of a nonsingular matrix?
A matrix A is said to be skew symmetric if AT = -A. For example,is skew symmetric since If A is an n Ã— n skew symmetric matrix and n is odd, show that A must be singular
Let A be a nonsingular n × n matrix with a nonzero cofactor Ann and set c = det(A) / Ann Show that if we subtract c from ann, then the resulting matrix will be singular.
Let x and y be elements of R3, and let z be the vector in R3 whose coordinates are defined byLet X = (x, x, y)T and Y = (x, y, y)T Show that xTz = det(X) = 0 and yTz = det(Y) = 0
Show that evaluating the determinant of an n Ã— n matrix by cofactors involves (n! - l) additions andmultiplications.
Show that the elimination method of computing the value of the determinant of an n × n matrix involves [n(n - 1)(2n - l)]/6 additions and [(n - 1)(n2 + n + 3)]/3 multiplications and divisions.
Let A be an n × n matrix and α a scalar. Show that det(αA) = αn det(A)
Let A be a nonsingular matrix. Show that det(A-1) = 1 / det(A)
Show that if E is an elementary matrix then ET is an elementary matrix of the same type as E.
For each of the following, compute (i) det(A), (ii) adj A, and (iii) A-1.(a) (b)
Show that if A is nonsingular then adj A is nonsingular and (adj A)-1 = det(A-1)A = adj A-1
Show that if A is singular then adj A is also singular.
Show that if det(A) = I then adj (adj A) = A
Suppose that Q is a matrix with the property Q-1 = QT. Show that qij = Qij / det(Q)
Let A be a nonsingular n × n matrix with n > 1. Show that det(adj A) = (det(A))n-1
det(A + B) = det(A) + det(B) In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the matrices are n × n.
det(A) = det(B) implies A = B. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the matrices are n × n.
det(Ak) = det(A)k In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the matrices are n × n.
A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the
If A and B are row equivalent matrices, then their determinants are equal. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the matrices are n ×
Let A and B be 3 × 3 matrices with det(A) = 4, det(B) = 6 and let E be an elementary matrix of type I. Determine the value of each of the following: (a) det(1/2A) (b) det(B−1AT) (c) det(EA2)
Let A be a matrix with integer entries. If | det(A) | = 1, then what can you conclude about the nature of the entries of A-1? Explain.
Given(a) Compute the value of det(A) (Your answer should be a function of x.) (b) For what values of x will the matrix be singular? Explain
Given the matrix(a) Compute the LU factorization of A. (b) Use the LU factorization to determine the value of det(A).
If A is a nonsingular n × n matrix, show that AT A is nonsingular and det(AT A) > 0.
Let A be an n × n matrix. Show that if B = S-1 AS for some nonsingular matrix S, then det(B) = det(A).
Let A and B be n × n matrices and let C = AB. Use determinants to show that if either A or B is singular, then C must be singular.
Let A be an n × n matrix and let λ be a scalar. Show that det(A - λI) = 0 if and only if Ax = λx for some x ≠ 0
Let x and y be vectors in Rn, n > 1. Show that if A = xyT then det(A) = 0.
Let x and y be distinct vectors in Rn (that is, x ≠ y) and let A be an n × n matrix with the property that Ax = Ay. Show that det(A) = 0.
Set A = round(10 * rand(6)). In each of the following, use MATLAB to compute a second matrix as indicated. State how the second matrix is related to A and compute the determinants of both matrices. How are the determinants related? (a) B = A; B(2, :) = A(1, :); B(1, :) = A(2, :) (b) C = A; C(3, :)
If a matrix is sensitive to roundoff errors, the computed value of its determinant may differ drastically from the exact value. For an example of this, set U = round(l00 * rand(l0)); U = triu((U, 1) + 0.1 * eye(10) In theory, det(U) = det(UT) = 10-10 and det(UUT) = det(U) det(UT) = 10-20 Compute
Use MATLAB to construct a matrix A by setting A = vander(1 : 6); A = A - diag(sum(A′)) (a) By construction, the entries in each row of A should all add up to zero. To check this, set x = ones(6, l) and use MATLAB to compute the product Ax. The matrix A should be singular. Why? Explain. Use the
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on 5 by α(x1, x2) = (αx1, αx1, αx2) (x1, x2) ⊕ (y1, y2) = (x1 + y1, 0) We use the symbol ⊕ to denote the addition operation for this system to avoid confusion with the usual addition x + y of
Let V denote the set of positive real numbers. Define the operation of scalar multiplication, denoted —¦, byÎ± —¦ x = xÎ±for each x ˆŠ R+ and for any real number a. Define the operation of addition, denoted Š•, byx Š• y = x ˆ™ y for all x, y ˆŠ R+Thus for this system the
Let Z denote the set of all integers with addition defined in the usual way and define the scalar multiplication, denoted ◦, by a ◦ k = [[α]] ∙ k for all k ∊ Z where [[α]] denotes the greatest integer less than or equal to α. For example, 2.25 ◦ 4 = [[2.25]] ∙ 4 = 2 ∙ 4 = 8 Show
Let S denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by α{an} = {αan} {an} + {bn} = {an + bn} Show that 5 is a vector space.
We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1 + a2x + . . . + anxn-1 ↔ (a1, . . . , an)T = a Show that if p ↔ a and q ↔ b, then (a) αp ↔ αa for any scalar a. (b) p + q ↔ a + b. [In general, two vector spaces are said to be isomorphic if their
Let C be the set of complex numbers. Define addition on C by (a + bi) + (c + di) = (a + c) + (b + d)i and define scalar multiplication by α(a + bi) = αa + αbi for all real numbers a. Show that C is a vector space with these operations.
Show that Rm×n, with the usual addition and scalar multiplication of matrices, satisfies the eight axioms of a vector space.
Show that C[a, b], with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space.
Let V be a vector space and let x ∊ V. Show that: (a) β0 = 0 for each scalar β. (b) If αx = 0, then either α = 0 or x = 0.
Given(a) Is x ˆŠ Span(x1, x2)? (b) Is y ˆŠ Span(x1, x2)? Prove your answers.
Let {x1, x2,..., xk} be a spanning set for a vector space V. (a) If we add an additional vector xk+1 to the set, will we still have a spanning set? Explain. (b) If we delete one of the vectors, say xk, from the set, will we still have a spanning set? Explain.
In R2Ã—2 letShow that E11, E12, E21, E22 span R2Ã—2.
Let S be the vector space of infinite sequences defined in Exercise 15 of Section 1. Let So be the set of {an} with the property an → 0 as n → ∞. Show that So is a subspace of S.
Prove that if S is a subspace of R1 either S = {0} or S = R1.
Let A be an n × n matrix. Prove that the following statements are equivalent.(a) N(A) = {0}.(b) A is nonsingular.(c) For each b ∊ Rn, the system Ax = b has a unique solution.
Let U and V be subspaces of a vector space W. Prove that their intersection U ∩ V is also a subspace of W.
Let S be the subspace of R2 spanned by e1 and let T be the subspace of R2 spanned by e2. Is S U T a subspace of R2? Explain.
Let U and V be subspaces of a vector space W. DefineU + V = {z | z = u + v where u ∊ U and v ∊ V}Show that U + V is a subspace of W.
Let S, T and U be subspaces of a vector space V. We can form new subspaces using the operations of ∩ and + defined in Exercises 18 and 20. When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that a(b + c) = ab +
Show that Cn[a, b] is a subspace of C[a. b].
Let A be a particular vector in R2×2. Determine whether the following are subspaces of R2×2. (a) S1 = {B ∊ R2×2 | AB = BA} (b) S2 = {B ∊ R2×2 | AB ≠ BA} (c) S3 = {B ∊ R2×2 | BA = 0} Determine whether the following are spanning sets for R2.
Given the functions 2x and |x|, show that (a) These two vectors are linearly independent in C[- 1, 1]. (b) The vectors are linearly dependent in C[0, 1].
Prove that any finite set of vectors that contains the zero vector must be linearly dependent
Let v1, v2 be two vectors in a vector space V. Show that v1 and v2 are linearly dependent if and only if one of the vectors is a scalar multiple of the other.
Prove that any nonempty subset of a linearly independent set of vectors {v1, ..., vn} is also linearly independent.

Showing 6600 - 6700 of 11883

First
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved