Homework Answers
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(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0...
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 (I) (2 mark) Find the characteristic polynomial of matrix A. (II) (1 mark) Find eigenvalues of the matrix A. III) (2 mark) Find a basis for the eigenspaces of matrix A. IV) (1 mark) What is the algebraic and geometric multiplicities of its eigenvalues. (V) (2...
2. [-12 Points) DETAILS LARLINALG8 7.2.005. Consider the following. -4 20 0 1 -3 A =...
2. [-12 Points) DETAILS LARLINALG8 7.2.005. Consider the following. -4 20 0 1 -3 A = 040 P= 04 0 4 0 2 1 2 2 (a) Verify that A is diagonalizable by computing p-AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (91, 12, 13)...
please solve both 3. [-12 Points] DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible)...
please solve both
3. [-12 Points] DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 8 -2 A= P= Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP = 1. [0/2 Points] DETAILS PREVIOUS ANSWERS LARLINALG8 7.2.001. Consider the following. -11 40 A= -27 (a) Verify that A is diagonalizable by computing p-1AP. -1 0 p-1AP = 10 3...
2. (-12 points) DETAILS LARLINALG8 7.2.005. Consider the following. Toi-3 A = 5 - 1 0...
2. (-12 points) DETAILS LARLINALG8 7.2.005. Consider the following. Toi-3 A = 5 - 1 0 0 1 0 -6 4 - 4 P= 04 1 2 0 2 (a) Verify that A is diagonalizable by computing p-1AP. p-1AP = It (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues....
step by step please Consider the following 2 -1 A = 0-2 -2 0 0 0,...
step by step please
Consider the following 2 -1 A = 0-2 -2 0 0 0, P- -1 0 1 -3 04 0 1 2 (a) Verify that A is diagonalizable by computing p-1AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (11, 12, 13) =(
Two n x n matrices A and B are called similar if there is an invertible...
Two n x n matrices A and B are called similar if there is an invertible matrix P such that B = P-AP. Show that two similar matrices enjoy the following properties. (a) They have the same determinant. (b) They have the same eigenvalues: specifically, show that if v is an eigenvector of A with eigenvalue 1, then P-lv is an eigenvector of B with eigenvalue l. (c) For any polynomial p(x), P(A) = 0 is equivalent to p(B) =...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P = -1 1 We know that A and B have the same characteristic polynomial and the same eigenvalues. Suppose that 2 is one of the common eigenvalues and x = [4 1] is a corresponding eigenvector of A. Which of the following is the eigenvector of B corresponding to 2 ? "f1-4-0-0-0-0-01-10 #8: Select
How to do Part 3? -- Find e^(At), the exponential of matrix A, where t ∈ ℝ is any real number.
How to do Part 3? -- Find e^(At), the exponential of matrix A,
where t ∈ ℝ is any real number.
Part 1: Finding Eigenpairs [10 10 5 10 -5 Find the eigenvalues λ,A2 and their corresponding eigenvectors vi , v2 of the matrix A- (a) Eigenvalues: 1,222.3 (b) Eigenvector for 21 you entered above: Vi = <-1/2,1> (c) Eigenvector for 22 you entered above: Part 2: Diagonalizability (d) Find a diagonal matrix D and an invertible matrix P D,...
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0...
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 a diagonal matrix D such that P-AP =D (VI) (2 marks) Find A107 by writing as linear combination of eigenvectors of A. (VII) (2 marks) Find a formula for Ak for all non-negative integers k. (Can k be a negative integer?) VIII) (1 mark) Use (VII)...
(2) (15 marks) Consider matrices 2 A= and B= 8 12 -2 3 b= = []...
(2) (15 marks) Consider matrices 2 A= and B= 8 12 -2 3 b= = [] (VI) (2 marks) Find A16 by writing 7 as linear combination of eigenvectors of A. (VII) (2 marks) Find a formula for Ak for all non-negative integers k. (Can k be a negative integer?) (VIII) (1 mark) Use (VII) to find Alº7 and compare it with what you found in (VI). (IX) (2 mark) Is A similar to B? If yes, find an invertible...
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 (I) (2 mark) Find the characteristic polynomial of matrix A. (II) (1 mark) Find eigenvalues of the matrix A. III) (2 mark) Find a basis for the eigenspaces of matrix A. IV) (1 mark) What is the algebraic and geometric multiplicities of its eigenvalues. (V) (2...
2. [-12 Points) DETAILS LARLINALG8 7.2.005. Consider the following. -4 20 0 1 -3 A = 040 P= 04 0 4 0 2 1 2 2 (a) Verify that A is diagonalizable by computing p-AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (91, 12, 13)...
please solve both
3. [-12 Points] DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 8 -2 A= P= Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP = 1. [0/2 Points] DETAILS PREVIOUS ANSWERS LARLINALG8 7.2.001. Consider the following. -11 40 A= -27 (a) Verify that A is diagonalizable by computing p-1AP. -1 0 p-1AP = 10 3...
2. (-12 points) DETAILS LARLINALG8 7.2.005. Consider the following. Toi-3 A = 5 - 1 0 0 1 0 -6 4 - 4 P= 04 1 2 0 2 (a) Verify that A is diagonalizable by computing p-1AP. p-1AP = It (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues....
step by step please
Consider the following 2 -1 A = 0-2 -2 0 0 0, P- -1 0 1 -3 04 0 1 2 (a) Verify that A is diagonalizable by computing p-1AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (11, 12, 13) =(
Two n x n matrices A and B are called similar if there is an invertible matrix P such that B = P-AP. Show that two similar matrices enjoy the following properties. (a) They have the same determinant. (b) They have the same eigenvalues: specifically, show that if v is an eigenvector of A with eigenvalue 1, then P-lv is an eigenvector of B with eigenvalue l. (c) For any polynomial p(x), P(A) = 0 is equivalent to p(B) =...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P = -1 1 We know that A and B have the same characteristic polynomial and the same eigenvalues. Suppose that 2 is one of the common eigenvalues and x = [4 1] is a corresponding eigenvector of A. Which of the following is the eigenvector of B corresponding to 2 ? "f1-4-0-0-0-0-01-10 #8: Select
How to do Part 3? -- Find e^(At), the exponential of matrix A,
where t ∈ ℝ is any real number.
Part 1: Finding Eigenpairs [10 10 5 10 -5 Find the eigenvalues λ,A2 and their corresponding eigenvectors vi , v2 of the matrix A- (a) Eigenvalues: 1,222.3 (b) Eigenvector for 21 you entered above: Vi = <-1/2,1> (c) Eigenvector for 22 you entered above: Part 2: Diagonalizability (d) Find a diagonal matrix D and an invertible matrix P D,...
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 a diagonal matrix D such that P-AP =D (VI) (2 marks) Find A107 by writing as linear combination of eigenvectors of A. (VII) (2 marks) Find a formula for Ak for all non-negative integers k. (Can k be a negative integer?) VIII) (1 mark) Use (VII)...
(2) (15 marks) Consider matrices 2 A= and B= 8 12 -2 3 b= = [] (VI) (2 marks) Find A16 by writing 7 as linear combination of eigenvectors of A. (VII) (2 marks) Find a formula for Ak for all non-negative integers k. (Can k be a negative integer?) (VIII) (1 mark) Use (VII) to find Alº7 and compare it with what you found in (VI). (IX) (2 mark) Is A similar to B? If yes, find an invertible...
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