Suppose that A and B are n x n matrices. Which of the following statements is(are) correct?Statements b and da)lf 0 is an eigenvalue of A, then A is invertibleO b)The zero vector is in the eigenspace of A associated with an eigenvalue >OStatements a, b and dO c)The matrix A and its transpose AT have different sets of eigenvalues.d)lf 4 is an eigenvalue of A, then A - 4I is not invertibleO Statements b, c and dAPreviousQuiz Suppose that A and B are n x n matrices. Which of the following statements is (are) false?O d) The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue na)lf A-7 is a factor of the characteristic of A, then 7 is an eigenvalue of AStatements b, d and eO Statements b, c and eO Statements a, b and eO Statements b and eO e) If two matrices have the same set of eigenvalues, then they are similarO c)The eigenvalue of the n x n identity matrix is 1 with algebraic multiplicity nO b)A matrix A can have more than n eigenvaluesQuiz saved at 1:00

Recall : If λ is an eigen value of A then there exist non zero v such that Av=λv.v is called the…

Answered: Suppose that A and B are n x n…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A be an n x n matrix.Which of the following statements is (are) true? Ⓒa) If Ax = Ax for some scalar A, the x is an eigenvector of A Statements b and d Statements a, b and d b)lf v₁ and v₂ are linearly independent eigenvectors, then they correspond to distinct eigenvalues Statements a, c and d Oc)An eigenspace of A is a null space of a certain matrix Statements cand d d)if A² is a zero matrix, then the only eigenvalue of A is 0
Consider the following matrix: 1 -6 -6 -6 -4 -2 -2 8 -6 0 0 12 -1 -3 -3 0 A = a) Find the distinct eigenvalues of A, their multiplicities, and the corresponding number of basic eigenvectors. Number of Distinct Eigenvalues: 1 Eigenvalue: 0 has multiplicity 1 and corresponding number of basic eigenvectors 1 b) Determine whether the matrix A is diagonalizable. Conclusion: Official Time: A is diagonalizable A is not diagonalizable SUBMIT AND MARK
Consider the following matrices A and B and vector b. D)What is the algebraic and geometric multiplicities of the eigenvalues of A. f) Find A10b by writing b as linear combination of eigenvectors of A.G) Find a formula for Ak for all non-negative integers k.H) Use (g) to find A10b and compare it with what you found in (VI).I) Is A similar to B? If yes, find an invertible matrix such that P−1AP = B.
From practice sheet I need the solution To b)
Consider the following matrix: A = ..comm 3 0 0 00 03 0 -5 0-12 -3 12 000-2 a) Find the distinct eigenvalues of A, their multiplicities, and the corresponding number of basic eigenvectors. Number of Distinct Eigenvalues: 1 Eigenvalue: 0 has multiplicity 1 and corresponding number of basic eigenvectors 1 b) Determine whether the matrix A is diagonalizable. Conclusion: A is diagonalizable Question 13 A is diagonalizable A is not diagonalizable sk dom A in the diagonalizablo matrix boloy and P-1AP-D for
Suppose that A and B are n x n matrices. If A and B are similar matrices, which of the following statements is(are) correct O b) det(A) equals det (B), provided they have the same main diagonal entries O a) matrices A and B have the same characteristic polynomial and same eigenvalues O d)lf A is similar to B, then B is similar to A as well O Statements a, b, c and d O Statements a, and d O Statements a, c and d OdA= P¹BP. for all invertible matrix P O Statements a, b and d
b. Which of the following statements are true? Select all that apply. A. The matrices A and AT have the same eigenvalues, counting multiplicities. B. The sum of two eigenvectors of a matrix A is also an eigenvector of A. | C. Each eigenvalue of A is also an eigenvalue of A?. D. Each eigenvector of A is also an eigenvector of A?. E. Each eigenvector of an invertible matrix A is also an eigenvector of A1.

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