Let be given the following linear systemx + 2y – 3z = 43.x – y+ 5z = 24.x + y + (a? – 14)z = a + 2where a E R is a parameter.1-Determine the augmented matrix A associated to linear system (1) and give a row-echelon form.2- Determine the value of the parameter a E R for which the system (1) has:a- no solutionb- exactly one solutionc- infinitely manyd- solve the system (1) in the case of infinitely many solutions.solutions0.

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Answered: Let be given the following linear…

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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Consider the following system of equations: 2x+y=z=8 3x - 2y + 3z = 23 3x - 2y z=7 where x, y, and z are the amounts of the three goods produced in the economy. This system of equations will give us the equilibrium output for each good. To find the solution to this system of equations, we will carry out the following steps: (a) Let A denote the matrix for the coefficients in the system and b denote the vector of the right-hand sides. Write down A and b and their dimensions. (b) Calculate the determinant det(A) of the coefficients matrix. Is this coefficients matrix singular or nonsingular? Can we find the inverse of this coefficients matrix? Explain your reasoning. (c) Consider the following matrix: C = det (A) 8 12 B α 1-9 -7 where a and 3 are parameters. If matrix C is the inverse of matrix A (C = A−¹), what are the values of a and B? -1 (d) Using the inverse of the coefficients matrix A-¹ that you calculated in the previous part, solve for the (unknown) variables x, y, and z.
Provide a coherent answer for items C and D.
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.
write the following systems in the matrix form X=-y X=0 X = 3x - 2y Ў=2y-x DX = x ) 9 = 5x+y y=-x y=x+y
Solve a simple system: 3x + 2y − z = 2x − y + 5z = −22x + y = 3 a. Matrix representation of system linear equation above into a matrix representation in form of Ax = B then find x= = A−1B, which represents x = [x y z] b. Cramer’s rule method to find x, y, and z.
Find the solutions (x1, x2, x3) for the linear system with various parameters (a, b). 2x13x2+ ax3 = 1 2x13x2+2x3 = b 4x13x2+2x3 = 4 (a) a = 4, Vb Є R (b) a = 2,b=1 (c) a=2, Vb ERǝb 1 Perform row reduction starting with the augmented matrix and ending with reduced row echelon form. Show each row reduction step. If the system has no solutions, explain why. If the system is consistent, find all solu- tions. In the case of infinitely many solutions, use parameters.
et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'? I would be very thankful if you could provide some explanation with the steps, thank you in advance.
1. 0 2 4 1 -1 10 0 0 1 1 The reduced row echelon form of the augmented matrix of a linear system is given above. Describe the solution set of this linear system in parametric vector form.

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