Let A be an 3 x 3 matrix with the characteristic equation det(113 - A) =...
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...
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
(3 points) Let A be a 4 x 4 matrix with det(A) = 8. 1. If the matrix B is obtained from mes the second row to the first, then det(B) = 2. If the matrix C is obtained from A by swapping the first and second rows , then det(C) = 3. If the matrix D is obtained from A by multiplying the first row by 5, then det(D) =
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
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,...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. --1:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of...
please do both 1 & 2 () There is interesting relationship2 between a matrix and its characteristic equation that we explore in this exercise. 2 (a) We first illustrate with an example. Let B - 1 -2 i. Show that 2-4 is the characteristic polynomial for B ii. Calculate B2. Then compute B2+ B 412. What do you get? (b) The first part of this exercise presents an example of a matrix that satisfies its own characteristic equation. Explain for...
Q1. Let A = be a 2 x 2 matrix. 30 (a) Find the characteristic polynomial of the matrix A. (5 pts) (b) Find all eigenvalues and associated eigenvectors of the matrix A. (10 pts) (c) If is an eigenvalue of A, what do you think it would be the eigenvalue of the matrix 7A?(Justify your answer) (5 pts)
Find the characteristic equation of A, the eigenvalues of A, and a basis for the eigenspace corresponding to each eigenvalue. A = Find the characteristic equation of A, the eigenvalues of A, and a basis for the eigenspace corresponding to each eigenvalue. -7 16 0 1 1 005 (a) the characteristic equation of A 2+7 2–1 2–5 = 0 (1 - 5)(1 - 1)(x + 7) = 0 (b) the eigenvalues of A (Enter your answers from smallest to largest.)...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the...