Given the matrix 5 28 -16 A = 1 8 -4 E R3x3, 3 21 -11 1. find all eigenvalues of A, 2. find the corresponding eigenvectors of A 3. show that A is diagonalizable, that is, find an invertible matrix KER3x3, and a diagonal matrix DE R3x3 such that 3. show that A is diagonalizable, that is, find an invertible matrix KER3x3, and a diagonal matrix DE R3x3 such that K-IAK = D.
Let 4-β 0 0 A=1 0 4-3 024-β where β > 0 is a parameter. (a) Find the eigenvalues of A (note the eigenvalues will be functions of β). (b) Determine the values of β for which the matrix A is positive definite. Determine the values of β for which the matrix A is positive semidefinite. (c) For each eigenvalue of A, find a basis for the corresponding eigenspace. (d) Find an orthonormal basis for R3 consisting of eigenvectors of...
Bycalculatingthecharacteristicpolynomial,eigenvaluesanddimensionsoftheeigenspaces of each map or matrix below, determine if the given map or matrix is diagonalizable. If a map or matrix is diagonalizable, diagonalize it (linear algebra) 1. By calculating the characteristic polynomial, eigenvalues and dimensions of the eigenspaces of each map or matrix below, determine if the given map or matrix is diagonalizable. If a map or matrix is diagonalizable, diagonalize it (that is, give a basis consisting of its eigenvectors) The field F over which you consider the...
(2) The matrix A-[-5 12 12 5 represents a reflection combined with a scaling by a factor of 13. (a) Find vectors ui and v2 such that A is reflection over the line L = span(n) and such that v2 is orthogonal to vi (b) Find the eigenvectors of A with their associated eigenvalues. (c) Find an eigenbasis B for A (a basis of R2 consisting of eigenvectors of A). (d) what is the matrix of the linear transformation T(z)...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
Question 1: Given the following matrix A. 02 A- 1 2 3 2 (a) Find the determinant of A (b) Find eigenvalues and the corresponding eigenspaces of A (c) Determine whether A is diagonalizable. If so, find a matrix P and a diagonal matrix D such that P-1AP=D If not, justify your answer. (d) Find a basis of Im(A) and find the rank of Im(A) (e) Find a basis of Ker(A) and find the rank of Ker(A) Question 1: Given...
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
Let matrix M = -8 -24 12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP−1. If not, explain carefully why not.
2. (12 pts) Given the matrix in a R R-E form: [1 1 0 0 3 0 0 0 1 0 -2 0 A = 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 (a) (6 pts) Find rank(A) and nullity(A), and nullity(AT). (b) (2 pts) Find a basis for the row space of A. (c) (2 pts) Find a basis for the column space of A. (d) (2 pts) Find a...
Q2. Consider the matrix A 6 3 0 -1 0-2 0 5 (a) Find all eigenvalues of the matrix A. (b) Find all eigenvectors of the matrix A. (c) Do you think that the set of the eigenvectors of A is a basis for the vector space R3? (Justify your answer