Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Expert Solution & Answer
Chapter 2, Problem 20RP
Explanation of Solution
a.
Unique solution:
- Suppose that the rank of an
Explanation of Solution
b.
Infinite number of solutions:
- Suppose that the rank of matrix
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
. The determinant of an n X n matrix can be used
in solving systems of linear equations, as well
as for other purposes. The determinant of A can
be defined in terms of minors and cofactors. The
minor of element aj is the determinant of the
(n – 1) X (n – 1) matrix obtained from A by
crossing out the elements in row i and column j;
denote this minor by Mj. The cofactor of element
aj, denoted by Cj. is defined by
Cy = (-1y**Mg
The determinant of A is computed by multiplying
all the elements in some fixed row of A by their
respective cofactors and summing the results. For
example, if the first row is used, then the determi-
nant of A is given by
Σ (α(CI)
k=1
Write a program that, when given n and the entries
in an n Xn array A as input, computes the deter-
minant of A. Use a recursive algorithm.
If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A.
A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) :
1
-1
-1
1
3
1
-3
1
-1
a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy.
b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy.
#UsePython
USING PYTHON
A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1).
Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added.
Please show steps and explain.
Chapter 2 Solutions
Introduction to mathematical programming
Ch. 2.1 - Prob. 1PCh. 2.1 - Prob. 2PCh. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.1 - Prob. 5PCh. 2.1 - Prob. 6PCh. 2.1 - Prob. 7PCh. 2.2 - Prob. 1PCh. 2.3 - Prob. 1PCh. 2.3 - Prob. 2P
Ch. 2.3 - Prob. 3PCh. 2.3 - Prob. 4PCh. 2.3 - Prob. 5PCh. 2.3 - Prob. 6PCh. 2.3 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Prob. 9PCh. 2.4 - Prob. 1PCh. 2.4 - Prob. 2PCh. 2.4 - Prob. 3PCh. 2.4 - Prob. 4PCh. 2.4 - Prob. 5PCh. 2.4 - Prob. 6PCh. 2.4 - Prob. 7PCh. 2.4 - Prob. 8PCh. 2.4 - Prob. 9PCh. 2.5 - Prob. 1PCh. 2.5 - Prob. 2PCh. 2.5 - Prob. 3PCh. 2.5 - Prob. 4PCh. 2.5 - Prob. 5PCh. 2.5 - Prob. 6PCh. 2.5 - Prob. 7PCh. 2.5 - Prob. 8PCh. 2.5 - Prob. 9PCh. 2.5 - Prob. 10PCh. 2.5 - Prob. 11PCh. 2.6 - Prob. 1PCh. 2.6 - Prob. 2PCh. 2.6 - Prob. 3PCh. 2.6 - Prob. 4PCh. 2 - Prob. 1RPCh. 2 - Prob. 2RPCh. 2 - Prob. 3RPCh. 2 - Prob. 4RPCh. 2 - Prob. 5RPCh. 2 - Prob. 6RPCh. 2 - Prob. 7RPCh. 2 - Prob. 8RPCh. 2 - Prob. 9RPCh. 2 - Prob. 10RPCh. 2 - Prob. 11RPCh. 2 - Prob. 12RPCh. 2 - Prob. 13RPCh. 2 - Prob. 14RPCh. 2 - Prob. 15RPCh. 2 - Prob. 16RPCh. 2 - Prob. 17RPCh. 2 - Prob. 18RPCh. 2 - Prob. 19RPCh. 2 - Prob. 20RPCh. 2 - Prob. 21RPCh. 2 - Prob. 22RP
Knowledge Booster
Similar questions
- Use a software program or a graphing utility with matrix capabilities to write v as a linear combination of u1, u2, U3, U4, and u5. Then verify your solution. (Enter your answer in terms of u1, u2, u3, U4, and u5.) v = (5, 2, -12, 13, 6) u1 = (1, 2, -3, 4, –1) u2 = (1, 2, 0, 2, 1) Uз 3D (0, 1, 1, 1, —4) U4 = (2, 1, -1, 2, 1) u5 = (0, 2, 2, -1, -1) V =arrow_forwardAssume A is k x n-matrix and B is n x l-matrix and AB = 0. Prove that rank(A) + rank(B) < n.arrow_forwardProgram the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)arrow_forward
- suppose a computer solves a 100x100 matrix using Gauss elimination with partial pivoting in 1 second, how long will it take to solve a 300x300 matrix using Gauss elimination with partial pivoting on the same computer? and if you have a limit of 100 seconds to solve a matrix of size (N x N) using Gauss elimination with partial pivoting, what is the largest N can you do? show all the steps of the solutionarrow_forwardFind the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n xn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable. Find the eigenvalues. (Enter your answers as a comma-separated list.) Is there a sufficient number to guarantee that the matrix is diagonalizable? O Yes O No Need Help? Read itarrow_forwardLet E : y2 = x3 + 3x + 4 be an elliptic curve over F37. 1, Find all the elements of the elliptice curve group 2, Find the order of the 3, Find a primitive element of this group and call it G. 4, Compute [30]G = G ⊕ G ⊕ · · ⊕ G (addition of 30 many G’s)arrow_forward
- Consider the following. 0 1 -3 A = 0 4 0 4 1 2 2 2 -1 2 (a) Verify that A is diagonalizable by computing P-AP. p-1AP = (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues.arrow_forwardUsing Python, to help solve the following problem. Provide an explanation of your solutions to the problem. 4. A symmetric matrix D is positive definite if x¹TDx > 0 for any nonzero vector x. It can be proved that any symmetric, positive definite matrix D can be factored in the form D = LLT for some lower triangular matrix L with nonzero diagonal elements. This is called the Cholesky factorization of D. Consider the matrix [2.25 -3 4.5 -10 -3 5 4.5 -10 34 a. Is A positive definite? Explain. A = b. Find a lower triangular matrix L such that LLT = A.arrow_forwardWrite a program that can inverse a matrix by using an approach to find its minor, cofactor and adjugate matrix. Test your program with the given matrix ?arrow_forward
- Pls Use Python If there is a non-singular matrix P such as P-1AP=D , matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. Ps: Please also explain step by step with " # "arrow_forwardAssignment.. تم الحفظ على الجهاز Q1: Construct a 3x3 nonzero matrix A such that the vector is a solution of Ax = 0. Q2: Delermine if the columns of the matrix form a limcarly independent set. Justify cach answer. 1-3 7 -1 -3 014arrow_forwardAssume A is k x n-matrix and B is n x l-matrix. Prove that rank(AB) < rank(A) and rank(AB) < rank(B)arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole