For each of the following parts, answer the question that is asked, and provide a brief explanation of your answer. Note that if you are claiming that something always happens then an example is not a sufficient justification, while if you are claiming that something sometimes happens then an example is enough. a) True or False: If A is ann x n matrix and the rows of A are linearly independent, then the columns of A2 must be linearly independent. b) True or False: There is a 2 x 2 matrix with characteristic polynomial X + 3X2 +3X+ 1. c) Suppose that A is an m x k matrix, and that CE) null(A) = span What is the dimension of col(A)? Note: For full credit your answer should be an actual number, not something inyolving m or k but if

For each of the following parts, answer the question that is asked, and provide a brief explanation of your answer. Note that if you are claiming that something always happens then an example is not a sufficient justification, while if you are claiming that something sometimes happens then an example is enough. a) True or False: If A is ann x n matrix and the rows of A are linearly independent, then the columns of A2 must be linearly independent. b) True or False: There is a 2 x 2 matrix with characteristic polynomial X + 3X2 +3X+ 1. c) Suppose that A is an m x k matrix, and that CE) null(A) = span What is the dimension of col(A)? Note: For full credit your answer should be an actual number, not something inyolving m or k but if

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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a. Suppose the columns of A are linearly independent. Given any b, what are the possible numbers of solutions for Ax = b? Justify your answer.
I thought that a matrix spanned Rn if there was a pivot in every row. However, when a matrix is linearly dependent and the last row is made up of zeroes, it could still end up potentially spanning Rn. I'm a little bit confused about how that works because there isn't a pivot in every row in that situation. How does it still span when there's a free variable? For context, I'm looking at Practice Problem 2 in Section 1.3 of the Vector Equations notes in the textbook Linear Algebra and Its Applications, Fifth Edition by David C. Lay, Steven R. Lay, and Judi J. McDonald. If a picture should be needed, here it is.
If A-1 = 3 3 -9 -12 , then the a2,1 entry of the matrix A is
A, B are two matrixes in size mxn, kxm respectively. Which of the claims are right? If the claim is right, prove it. If the claim isn't right, bring a counter example. a if the first two rows of A are equal, then the first two rows of BA are equal too. b) if the first two columns of A are equal, then the first two columns of BA are equal too. c) if there is no Zeroes in the main diagonal of A₂ then there is no Zeroes in the main diagonal BA too. of KRE
In Exercises 11 and 12, the matrices are all n × n. Each part of the exercises is an implication of the form "If "statement 1”, then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever “statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Justify each answer. 11. a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n - n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R". d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions. e. If AT is not invertible, then A is not invertible.
8 - Which of the following statements is FALSE? a) Let A and B be 3 x 3 matrices. If det(A) = 3 and det(B) = 4, then det(AB) = 12. b) Let A be a 4 x 4 matrix. If det(A) = 2 then det(A) =. c) Let A and B be 2 x 2 matrices. Then, AB = BA. d) If det(A) = 1 and det (2A) = 16, then A is a 4 x 4 matrix. e)None
Read the following definition carefully, and then answer the questions below. i We define an n x n matrix A to be passive if A? = A. Note: before diving in to solving the questions below, you may wish to try to create some examples of passive mat rices (say 2 × 2 to keep it simple). 21 Suppose that A is passive. Show that I – A is also passive but A – I is not passive. 22 True or False:' if A is m × n and B is n x m and AB = Im, then BA is passive. 23 Show that if A is a passive 2 x 2 matrix with non-zero entries, then the sum of the entries on the diagonal of A is equal to 1. Create an example of such a matrix and show that it is indeed passive. (That is, if A : Ja b is passive and has a, b, c, d # 0, then a+d = 1.) 2.
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Parts A and B please
Executing the following sage code constructs a matrix Q, then prints the matrix and its reduced row echelon form. Q=matrix(QQ, 4, 3, [-7, -22,-4, 12, 137,-13,-18, -213,21,15,115,-5]) print(Q) print() print(Q.rref()) The output is: -4] [ -7 -22 [ 12 137 -13] [-18 -213 21] [ 15 115 -5] [ [ [ 1 0 0 0 6/5] 1 -1/5] 0 0 0] 0] Are the columns of matrix Q linearly dependent, or linearly independent?
The table below shows the number of tickets sold for seats in the stalls (S), circle (C) and upper circle (U) of a theatre for a show on Monday night and Tuesday night. (a) S 285 (c) Monday с 268 U S 182 279 Tuesday с 275 U 177 Represent this information in the 2 x 3 matrix A, so that A₁₁ is the number of stalls tickets sold for Monday and A2,3 is the number of upper circle tickets sold for Tuesday. Ticket prices to sit in the stalls, circle and upper circle were $88, $119 and $78 respectively. (b) Represent the ticket price information in matrix B so that matrices A and B can be multiplied together to create a meaningful result. State the order in which matrices A and B must be multiplied together, and determine the result of this multiplication, matrix M. (d) State the night on which the value of ticket sales was highest and calculate by how much the sales on this night exceeded the other night.