Given the matrix:A42-1 0A-3 is an eigenvalue of A and (-1, 1, 1) is a corresponding eigenvector. A = -2 is an eigenvalue ofA of multiplicity 2.(a) What is the characteristic polynomial of A?(b) Find the eigenvector(s) corresponding to λ = -2.(c) Find the general solution of x' = Ax.

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Answered: Given the matrix: A 4 2 -1 0 A-3 is an…

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 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0) B. (a 0/2 -5) C. (-5 2/0 a) D. (a -5/0 2) E. (a 2/0 -5)
The matrix A := C = 12 -2 -3 13 has an eigenvector 6 (8) with eigenvalue 10. What is c?
Suppose A is a 3 x 3 matrix with eigenvectors [1] V1 = V2 = 1 and v3 = corresponding to eigenvalues A1 =-, d2 = }, and A3 = 1, respectively. Find A and A20.
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
A H the eigenvalues X₁ = 2 and A₂ = -1, respectively. What is A(v₁ + V₂)? Suppose A is a 3 x 3 matrix with eigenvectors V₁ A 3 2 B Ⓡ = 0 and V₂ = 2 -2 corresponding to D H
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.
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
[1 2. (a) Find the eigenvalues and eigenvectors of B = (b) Explain why the matrix A cannot be diagonalized.
(a) Find the eigenvalues and eigenvectors of the matrix A. Call them A1 and A2 and the corresponding eigenvectors v(1) and v(2). (b) Show that l2 = 1; and v2) = v(1)*, where the * indicates complex conjugation. %3D
Matrix A has the eigenvalue 2 = 5 and eigenvector v. Find generalized eigenvector. 7 1 01 1|,λ5, ν = A = -3 4 -2 1 41 X2=

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Given the matrix: A 4 2 -1 0 A-3 is an eigenvalue of A and (-1, 1, 1) is a corresponding eigenvector. A = -2 is an eigenvalue of A of multiplicity 2. (a) What is the characteristic polynomial of A? (b) Find the eigenvector(s) corresponding to λ = -2. (c) Find the general solution of x' = Ax.