use the definition of the characteristic polynomial for now on. χA(λ) = det(λI − A).
use the definition of the characteristic polynomial for now on. χA(λ) = det(λI − A). 1....
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
1. Consider the matrix (a) Find the characteristic polynomial and eigenvalues of A (b) Find a basis for the eigenspace corresponding to each eigenvalue of A. (c) Find a diagonalization of A. That is, find an invertible matrix P and a diagonal matrix such that A - POP! (d) Use your diagonalization of A to compute A'. Simplify your answer.
Differention Equations - Can someone answer the checked numbers please? Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
11 (10) [10pts) Let A be the 2 x 2 matrix A-L 5 A = 2 4 The characteristic polynomial of A is (3- A) (1 - ). Compute an invertible matrix P and diagonal matrix D such that A= PDP-
Chapter 5, Section 5.1, Question 24a Find det (A) given that A has p(A) as its characteristic polynomial. det (A) = Hint: See the proof of Theorem 7.1.4. ( If given det (ÀI-A) = λη + C1A + … + cn then, on setting λ 0, det (-A) = cn or (-1)ndet (A) = cn ) Click if you would like to Show Work for this question: Open Show Work n -1 Chapter 5, Section 5.1, Question 24a Find det...
3. (a) For the following matrix A, compute the characteristic polynomial C(A) = det(A ?): A-1 1 (b) Find all eigenvalues of A, using the following additional information: This miatrix has exactly 2 eigenvalues. We denote these ??,A2, where ?1 < ?2. . Each Xi is an integer, and satisfies-2 < ?? 2. (c) Given an eigenvalue ?? of A, we define the corresponding eigenspace to be the nullspace of A-?,I; note that this consists of all eigenvectors corresponding to...
1-11 23 )--[-!?). - (111) DE 1 0 0 4 1 - 4 4 0-3 0 0 0 3 0 0 -1 0 5 4 2-3 E = 6. Consider the matrix A, above. Use diagonalization to evaluate A. 7. Consider the matrix B, above. Find a diagonal matrix D, and invertible matrix P, such that B = PDP- 8. Consider the matrix C, above. Find a diagonal matrix D, and invertible matrix P, such that C = PDP-!. If...
Answer 7,8,9 1-11-1)--[-13.-(41-44)--:-- 3 1 0 0 -1 0 5 4 2-3 0 0 0 6. Consider the matrix A, above. Use diagonalization to evaluate A. 7. Consider the matrix B, above. Find a diagonal matrix D, and invertible matrix P, such that BPDP-1 8. Consider the matrix C, above. Find a diagonal matrix D, and invertible matrix P, such that C = PDP-1. If this is not possible, thus the matrix is not diagonalizable, explain why. 9. Consider the...
(1 point) Find the characteristic polynomial of the matrix. Use x as your variable instead of λ. -3 2