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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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.
Let A and B be nxn matrices. Show that if A is an eigenvalue of AB, then A is also an eigenvalue of BA.
Let A be an 3 by 3 matrix. Select all true statements below. A. If A is diagonalizable, then A has 3 distinct real eigenvalues. B. If A has 3 linearly independent eigenvectors, then A is diagonalizable. C. The matrix A may or may not be diagonalizable. D. The matrix A is certainly diagonalizable. E. If A has 3 distinct real eigenvalues, then A is diagonalizable. OF. If A is diagonalizable, then A has 3 linearly independent eigenvectors. G. None of the above
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
Let A be an n x n matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to A, together with the zero vector, is called the eigenspace of A and is denoted by EA. qs)Why is it important that this definition includes the phrase "together with the zero vector"? (i.e., what would go wrong if this phrase was not included?)
ich of the following statements is not true? a) An n x n matrix that is orthogonally diagonalizable must be symmetric. b) For any matrix A, A" A is symmetric. c) If A is symmetric, any two eigenvectors of A corresponding to different eigenvalues must be orthogonal. d) If A and B are symmetric matrices, AB must also be symmetric.
i. Show that the matrix is not diagonalizable. ii. Find the eigenvalues of the matrix and determine whether there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable. -3 -2 3 3 4 -9 1. 2 -5)
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.
Let A be an invertible 3x3 matrix. Mark all that are necessarily true about this matrix: The dimension of the vector space spanned by the columns of A is at most 2. The columns of A are linearly independent. O A is diagonalizable O O is not an eigenvalue of A.
-1 Question6Let A =|1 3 a) Find all eigenvalues and corresponding eigenvectors. b) Find matrices P and D such that P is nonsingular and D = P-1AP is diagonal. c) Compute A3.

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