Question 3: Consider the integer programming problemmin (c, x)s.t. Ax > b,x > 0,(1)x – integer,in which the matrix A has integer entries. Prove that its optimal value is not smallerthan the optimal value of the linear programming problemmax ([6], X)s.t. A"A < c,1> 0.The symbol [6] denotes the roundup of the vector b: the smallest integer vectorgreater than or equal to b.

Consider the LPP: minc, xs.t.Ax≥b,x≥0, x-integer . . . . (1) A also has integer…

Answered: Question 3: Consider the integer…

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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1. (a) Is there any 4 x 7 matrix A that satisfies Nullity(A) = 2. If yes, find an example of such A; if not, explain your answer. (b) Is there any 10 x 5 matrix B that satisfies Rank(B) = 7. If yes, find an example of such B; if not, explain your answer. (c) Is there any 5 x 6 matrix C that satisfies Nullity(C) = 2. If yes, find an example of such C; if not, explain your answer.
Solve the following Linear Programming Problem using the two-phase method, showing all the interactions:
5. Suppose the matrix A is 27 × 36 and the rank of A is 14. Find the nullity of A and the dimension of each of the following spaces: C(A), N(A), R(A), and L(A).
Solve the following system using the Gaussian algorithm: 2x1 - x2 + 3x3 = 0 -2x1 + x2 - X3 = -2 x1 - 3x2 + 6x3 = 3 Let x be the vector x = |x2, where x1, x2 and x3 are solutions of the system, and y be the vector y =|1 Evaluate the scalar product of x and y. [Please enter your answer numerically in decimal format. You will be marked correct as long as what you enter is within 0.25 of the correct answer. So for example, if the correct answer is 6.78 then any input that lies between between 6.53 and 7.03 will be marked as correct.] Answer:
8. (i)Redo: For what type of square matrices A will the algebraic system Ax = 2x have solutions other than x = 0? { Here x is an nx1 real matrix & A is an nxn real matrix.} Hints: 1) Ix = x. 2) Use GP appropriately; 3) Such matrices A must satisfy a certain relation.
After 25 years, your savings account has now grown to $16,200 and has an interest of 6.1% that compounds continuously. How much did you deposit at the beginning of the account
Given the system Ax b below: 1 2 4 1 X = 1 -1 5 2 3 (a) Find the Least Squares solution x of this system. [Hint: if you get awful fractions, check your work!] (b) Find the projection p of b onto the column space of A. [Hint: use x.] (c) Find the residual r = b – p. (d) Verify thatre N(A"), i.e., that the residual is in the nullspace of the transpose of A. (e) Did you actually solve the given 4 x 2 system? If not, what 4 x 2 system did you solve? Explain briefly.
Question 4. Suppose your linear code C € V(10, 11) is given by [x1,x2,...,x10] € C if x₁ + x10 = 0, + x₁ + 2x₂ + (a) Find a parity check matrix. + 10x10 = 0.

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