1. Consider1 4A =2 2a4||1a2bIf A is an augmented matrix of a system of linear equations, find the value(s) of aand b such that the systemis inconsistenti)ii)iii)has unique solutionhas infinitely many solutionsCO

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Answered: 1. Consider [1 4 A = 2 2a 4 1 a 2b If A…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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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…
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Suppose that the following matrix A is the augmented matrix for a system of linear equations. ... 1 2 3 4 -2 a? -7 -11 A = 2 -1 -1 a where a is a real number. What is the sum of all values of a that will make the system consistent (it has at least one solution)? А) о 4 -1 1
1. By using back-substitution to solve the systems a) x - y = 4, 2y + z = 6, 3z = 6; b) -x + y – z = 0, y +z = 6, 2z = 4; 2. Solve the system of linear equation x + y +z = 6, 2x – y + z = 3, 3x – z = 0; 3. Determine the values of k such that the following system of linear equation does not have a unique solution x+y+ kz = 3, x+ ky +z = 2, kx +y +z = 1; 4. Show that if ax? + bx +c= 0 for all x, then a = 6 = c= 0; -1 5. Find the product of matrices A = 3 and B = 1. -2 2 -1 -3 -2 6. Given the matrices -1 1 1. -2 2 3 A = B = 3. 4 -1 5 1 Find 3A – 2B; 7. Given a square matrix 1 -2 A = -3 -1 4 1 Find A? + 2A; 8. Solve the system using Gaussian elimination with back-substitution x + 2y = 0, x + y = 6, 3x – 2y = 8; 9. Solve the system using Gauss-Jourdan elimination x – 3z = -2, 3x + y – 2z = 5, 2x + 2y + z = 4; 10. Find the solution set of the system of linear equations represented by the following augmented matrix 1 -1 3 1 -2 1 1 -1
5. Choose the following statements are True (T) or False (F): (a). det(A + B) = det (A) + det (B) (b). A system of linear equations is consistent if it has at least one solution (c). The transpose of a upper triangular matrix is a lower triangular matrix (T) or (F) (T) or (F). (T) or (F).
Suppose that the augmented matrix of a system of linear equations for unknowns x, y, and z is 16 13 3 2 3 -12 16 ! -4 -4 16 The system has: Solve the system and provide the information requested. O a unique solution which is 30 O infinitely many solutions Ono solution Continue F1 64 3 two of which are @ 2 47 2 F2 31 <= 0 = 0 = 0 x = 0 x = 0 #3 y = y = 80 F3 0 2 = 7 $ F4 % 5 0/0 9 X F5 A 6 F6 & 7 F7 2023 Mc
2. Suppose that the rref of an augmented matrix of a linear system results as 1 0 1 13 0 1 0 21 000 00 (a) What are the leading variables? (b) What are the free variables? (c) What is the corresponding system of equations in terms of the free variables. (d) Write the solutions in vector form.
3x 2 1-2 0 0 3 --(3) -- 67 9) --C) --(2) (129) A == B 0 D = 5 -3 -1 -4 -2 3 Where possible, find the following. If not possible give the reason. (a) AB (b) DA (c) B+C (d) 2D + BC (e) AC (f) What is the value of x for which A does not have an inverse.

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1. Consider [1 4 A = 2 2a 4 1 a 2b If A is an augmented matrix of a system of linear equations, find the value(s) of a and b such that the system is inconsistent i) ii) ii) has unique solution has infinitely many solutions CO