Question 6. (A) Rewrite the system below in matrix form and then (B) determine whetheror not the given vector function satisfies the system.y₁ = y₁ + y₂y₂ = -2y₁ +4y2y₁ (0) = 1₂ (0) = 0= 2 [1] e² - [1¹] e³t3ty = 2

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Answered: Question 6. (A) Rewrite the system…

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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A=1 B-2 C=9 D=3
((((())))))
(a) (b) Consider the following two systems. (1) Find the inverse of the (common) coefficient matrix of the two systems. A-¹ = y = y = -3x - 4y -4x + 3y -3x - 4y -4x + 3y = = = = -2 -2 2 -3 (ii) Find the solutions to the two systems by using the inverse, i.e. by evaluating A ¹B where B represents the right hand side (i.e. B = Solution to system (a): x = Solution to system (b): for system (a) and B = [2] for system (b)).
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.
(c) Consider the system 2х1 — бх, — Хз — 38 — Зх1 — х2 + 7Хз :-34 -8x1 + x2 – 2x3 = -20 (i) Write the system in AX = B. 1 0 | and 23/10 1. (ii) Given the decomposition of matrix A is L= -3/2 -4 -6 -1 -43 b -11 -5 a 1 U = (0 -10 and the Al: 11/2 -373/20, -62 373 -11 C -20/ Find a, b and c.
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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Question 6. (A) Rewrite the system below in matrix form and then (B) determine whether or not the given vector function satisfies the system. y₁ = y₁ + y₂ y₂ = -2y₁ +4y2 y₁ (0) = 1 ₂ (0) = 0 = 2 [1] e² - [1¹] e³t 3t y = 2