Problem 2Consider the matrix A =-3(a) Find the eigenvalues A₁, A2 of A. (X₁ < A₂)A₁ =A₂ =cont.(b) Find the corresponding normalized eigenvectros ₁ =(9)x₂ =a=b =C=

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Answered: Problem 2 10 -3 Consider the matrix 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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0.2 0.3 Find eigenvalues ())) B dy -2 eigenvectors. df = [²3²] dt 3-2
Question 7 Find all eigenvalues and eigenspaces for the following matrices. (a) A = (b) 1 -1 = -3 3 B = 0 -5 3 0 0 4
35. (a) Show that the eigenvalues of the 2 X 2 matrix a b A = c d] с а are the solutions of the quadratic equation 12 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = {(a + d ± V(a – d)? + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) - λι + λ and det A = λιλε where A, and A2 are the eigenvalues of A. 36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
Consider the following initial-value problem. (;小) -1 -2 X' = X(0) = + (:)- Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) = Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenvalue. K1 K, = Solve the initial-value problem. X(t) =
Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1 and [1] an H Let x(t) be the position of a particle at time t. Suppose we solved the initial corresponding eigenvectors V₁ = value problem x = Ax, x(0) = [3] -ce + c₂e-t x(t) = = [2(t)] = [ qe * +ge + -3t Search and C2= -3t and V2 = to obtain then C1=
[5 -6 31 4. For matrix 6 -7 3 one of the eigenvalues equals 2. Find corresponding eigenvector. L6 -6 2]
please solve question (b)
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,

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Problem 2 10 -3 Consider the matrix A (¹92³). -3 (a) Find the eigenvalues A₁, A2 of A. (A1 < A2) A₁ cont. (b) Find the corresponding normalized eigenvectros ₁ = E 11 = b = C=L d= (8)