Question

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 eigenvalue ??, as well as O. For each eigenvalue of A, find the dimension of the corresponding eigenspace.

Answer 1

a. -lambda (- lambda - 1) (- lambda + 1) + 2 lambda - 2=0

b. lambda 1= - 2 and lambda 1=1

c. Eigen space corresponding lambda 1 has dimension 1, as the basis is consists of single vector (-1/3,-1,1)

And for lambda 2 the dimension is 1 as the basis consists only (1,0,0)