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Suppose A is a 3 x 3 matrix that is nilpotent but not zero. So Ak-0...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F)...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
مل 3 (1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3...
مل 3 (1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3 with corresponding eigenvector and an eigenvalue -1 with corresponding eigenvector Find an invertible matrix P and a diagonal matrix D so that A = PDP-1. Enter your answer as an equation of the form A = PDP-1. You must enter a number in every answer blank for the answer evaluator to work properly. 1-1
(1 point) Let 3 -4 A = -4 -1 -4 -2 -2 If possible, find an...
(1 point) Let 3 -4 A = -4 -1 -4 -2 -2 If possible, find an invertible matrix P so that D = P-1 AP is a diagonal matrix. If it is not possible, enter the identity matrix for P and the matrix A for D. You must enter a number in every answer blank for the answer evaluator to work properly. P= II II D= Be sure you can explain why or why Is A diagonalizable over R? diagonalizable...
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an...
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an eigenvector of Mk where k is any positive integer? If so, what would the associated eigenvalue be? Now suppose that the matrix A is nilpotent, i.e. A* integer k 2. Show that 0 is the only eigenvalue of A. [Hint: what is det (A)? This should help you decide that A has an eigenvalue of 0 in particular. Then you need to demonstrate that...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find...
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).
Review 4: question 1 Let A be an n x n matrix. Which of the below...
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)....
Let A be a diagonalizable n x n matrix and let P be an invertible n...
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = P-1AP is the diagonal form of A. Prove that Ak = Pekp-1, where k is a positive integer. Use the result above to find the indicated power of A. 0-2 02-2 3 0 -3 ,45 A5 = 11
4. (-12 points) DETAILS LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix...
4. (-12 points) DETAILS LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) -2 -2 A 0 3-2 0 -1 PE 11 Verify that p-IAP is a diagonal matrix with the eigenvalues on the main diagonal. P-AP Need Help? Read it Talk to a Tutor Submit Answer 5. [-12 Points] DETAILS LARLINALG8 7.2.013. For the matrix A, find (if possible) a nonsingular matrix P such that...
1 1 3 3 5. Diagonalize the matrix A = -3 -5 -3 if possible. That...
1 1 3 3 5. Diagonalize the matrix A = -3 -5 -3 if possible. That is, find an invertible matrix P and 3 3 a diagonal matrix D such that A = PDP-1 6. If u is an eigenvector of an invertible matrix A corresponding to , show that is also an eigenvector of A-!. What is the corresponding eigenvalue?
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
مل 3 (1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3 with corresponding eigenvector and an eigenvalue -1 with corresponding eigenvector Find an invertible matrix P and a diagonal matrix D so that A = PDP-1. Enter your answer as an equation of the form A = PDP-1. You must enter a number in every answer blank for the answer evaluator to work properly. 1-1
(1 point) Let 3 -4 A = -4 -1 -4 -2 -2 If possible, find an invertible matrix P so that D = P-1 AP is a diagonal matrix. If it is not possible, enter the identity matrix for P and the matrix A for D. You must enter a number in every answer blank for the answer evaluator to work properly. P= II II D= Be sure you can explain why or why Is A diagonalizable over R? diagonalizable...
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an eigenvector of Mk where k is any positive integer? If so, what would the associated eigenvalue be? Now suppose that the matrix A is nilpotent, i.e. A* integer k 2. Show that 0 is the only eigenvalue of A. [Hint: what is det (A)? This should help you decide that A has an eigenvalue of 0 in particular. Then you need to demonstrate that...
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).
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)....
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = P-1AP is the diagonal form of A. Prove that Ak = Pekp-1, where k is a positive integer. Use the result above to find the indicated power of A. 0-2 02-2 3 0 -3 ,45 A5 = 11
4. (-12 points) DETAILS LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) -2 -2 A 0 3-2 0 -1 PE 11 Verify that p-IAP is a diagonal matrix with the eigenvalues on the main diagonal. P-AP Need Help? Read it Talk to a Tutor Submit Answer 5. [-12 Points] DETAILS LARLINALG8 7.2.013. For the matrix A, find (if possible) a nonsingular matrix P such that...
1 1 3 3 5. Diagonalize the matrix A = -3 -5 -3 if possible. That is, find an invertible matrix P and 3 3 a diagonal matrix D such that A = PDP-1 6. If u is an eigenvector of an invertible matrix A corresponding to , show that is also an eigenvector of A-!. What is the corresponding eigenvalue?
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