The general solution for the linear homogeneous system depends only on the freevariables and there is a solution for any choice of these variables. The vectorcoefficients of each one of the free variables are called the fundamental solutions.If coefficient matrix A has dimensions 9x27, is the statement that the linear systemAx = 0 must have at least 18 fundamental solutions correct?TrueO False

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Answered: The general solution for the linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Select the correct statements from below for a linear system of equations Ax = b. The determinant of a permutation matrix is negative if it leads to an odd number of row interchanges in Gaussian elimination. A zero in the pivot position upon pivoting implies that the matrix A is singular. If A is a lower triangular matrix then the system could be solved by back-substitution. The number of floating point operations in back-substitution is of order O(n³). If the LU decomposition for A is given then the systems could be solved in O(n²) operation.
If a 6 x 6 matrix contains 19 zeros, show that there are 4 zeros which are all in distinct rows and columns.
Is it possible that all solutions of a homogeneous system of twelve linear equations in fourteen variables are multiples of one fixed nonzero solution? Discuss. Consider the system as Ax = 0, where A is a 12x 14 matrix. Choose the correct answer below. O A. Yes. Since A has at most 12 pivot positions, rank As 12. By the Rank Theorem, dim Nul A = 14- rank Az 2. Since there is at least one free variable in the system, all solutions are multiples of one fixed nonzero solution. O B. No. Since A has 12 pivot positions, rank A= 12. By the Rank Theorem, dim Nul A = 12 - rank A= 0. Since Nul A = 0, it is impossible to find a single vector in Nul A that spans Nul A. OC. No. Since A has at most 12 pivot positions, rank As 12. By the Rank Theorem, Nul A= 14 - rank Az 2. Thus, it is impossible to find a single vector in Nul A that spans Nul A. O D. Yes. Since A has 12 pivot positions, rank A= 12. By the Rank Theorem, dim Nul A= 12- rank A= 0. Thus, all solutions are multiples of one fixed nonzero…
Given:2a + 5b + 3c = 23a + b - 2c = 3-a + 2b + c = 1Identify the coefficient matri A and the augmented matrix A'of the given linear system.
Do 2b)
Answer the question
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4) Consider the system of linear equations: X1 + 5x2 + 3x3 = 4 X1 – X2 + 6x3 = 16 2x1 + x2 = 5 a) Write the system as Ax=b b) Solve the system using Gaussian Elimination with backward substitution. c) What is the determinant of A. d) Perform one step of the Gauss-Seidel Method on this system starting with initial guess 1 1
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1. A system of linear equations can be described in a matrix equation Ax = b with 2 -1 1] A = |1 1 2,x = |y| and b = |b2 [-1 5 4] a. Determine if the columns of the matrix A span R³. Give your arguments. b. If the system is homogeneous, describe the trivial solution of the system (if any) in parametric vector form. Also, give a geometric description of the solution set. Give the clear argument. c. If the system is nonhomogeneous and consistent, describe b in parametric vector form. Give the clear argument. d. If b, = 5, b2 = -3, bz = 0, is the system consistent? Give valid argument. Find the solution set of the system. Compare the solution set with the trivial solution of the homogeneous system in b.
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