True or false. Please justify why true or why false also
True or false. Please justify why true or why false also (I) A square matrix with...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no credit. (1) A square matrix with the characteristic polynomial X - 413 +212 - +3 is invertible. [23] (II) Matrix in Zs has two distinct eigenvalues. (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices Dand D2 are...
True or False? Justify your answer. Answers without correct justification will receive no credit. 1. Similar matrices have the same eigenspaces for the corresponding eigenvalues. 2. There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geometric multiplicity is 3.
True or False? Justify your answer. Answers without correct justification will receive no credit. (I) A square matrix with the characteristic polynomial λ 4 −4λ 3 + 2λ 2 −λ+ 3 is invertible. (II) Matrix 2 3 14 in Z5 has two distinct eigenvalues.
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 Compute and completely factor the characteristic polynomial of the following matrix: 0 A= -4 5 0 1 1 For credit, you have to factor the polynomial and show work for each step. B In the following, use complex numbers if necessary. For each of the following matrices: • compute the characteristic polynomial; • list all the eigenvalues (possibly complex) with their algebraic multiplicity; • for each eigenvalue, find a basis (possibly complex) of the corresponding eigenspace, and write the...
Part A. (True/False Questions) (15 pts). Decide if the given statement is true or false. (Justify briefly your answer) 1. The eigenvalues of the matrix A = -5 6 are: 5 and -4. O True False 2. Let A= 2 -4 be a square matrix. The vector v= [ is an eigenvector of the matrix A. 2 True False 3. If I = -4 is an eigenvalue of a 5 x 5 matrix A, then Av = -4v for any...
Let A be an n x n matrix. Then we know the following facts: 1) IfR" has a basis of eigenvectors corresponding to the matrix A, then we can factor the matrix as A = PDP-1 2) If ) is an eigenvalue with algebraic multiplicity equal to k > 1, then the dimension of the A-eigenspace is less than or equal to k. Then if the n x n matrix A has n distinct eigenvalues it can always be factored...
True or False? Justify your answer. Answers without correct justification will receive no credit. Two diagonal matrices D1 and D2 are similar if and only if D1 = D2.
Determine the following statements true or false (1) A linear operator A ∈ L(V) is similar to a diagonal matrix with eigenvalues on the diagonal if A is invertible. (2) Let A ∈ L(V). Then V = ελ1+...+ελk where λ1, ... ,λk are all distinct eigenvalues of A (3) Let A ∈ L(V). and λ be an eigenvalue of A. Then its eigenspace ελ is a subspace of its generalized eigenspace gελ
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).