?????Are the following statements true or false for a square matrix A?1. If an n x n matrix A is diagonalizable, then A has n distinct eigenvalues.2. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy.3. The eigenvalues of a matrix are the entries on its main diagonal.4. If u and v are linearly independent eigenvectors, then they correspond to distinct eigenvalues.5. A matrix A is singular if and only if 0 is an eigenvalue of A.

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Answered: Are the following statements true or…

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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4. True or False? If true, prove. If false, provide a counterexample. a. If same size matrices A and B have exactly the same eigenvalues (including multiplicities), they must be similar. b. If a square matrix A is diagonalizable, so is Ak, for any positive integer k. C. If a square invertible matrix A is diagonalizable, so is A-¹. d. If a square matrix A is invertible, then it's diagonalizable. e. For any m x n matrix A, the matrix AT A is orthogonally diagonalizable. f. Let A be a symmetric n x n matrix. Suppose ₁ and ₂ are eigenvectors corresponding to two distinct eigenvalues of A. Then, ||x₁ + x₂ || = ||X₁|| + ||x₂||. g. If an n x n matrix A has only real eigenvalues, then it must be symmetric.
Let A be an n xn real matrix with det( A) + 0. Which of the following statements is false? Select one: O a. All eigenvalues of A are nonzero. O b. The inverse matrix A-1 exists. O c. The rank of A is n. O d. There exist infinitely many solutions to the system Ax = b, for any vector b € R". %3D
True/False along with reasons and explanations- A is an nxn matrix that has no inverse, so the only eigenvalues of A is 0.- If the vector v is an eigenvector of the matrix A, then kv is also an eigenvector provided that k ≠ 0.- A is row equivalent to B, then A and B have the same eigenvalues.- It is known that the two eigenvectors are linearly independent, it can be concluded that they come from different eigenspaces.- Every matrix that has an inverse can be diagonalized.- If A is a non-singular matrix that can be diagonalized, then A^T and A^-1 can also be diagonalized.- A matrix of order 3x3. There is a basis R^3 which consists of the eigenvectors of A. Then A can be diagonalized.- If the matrix can be diagonalized, then the matrix is diagonally single.- A and B are similar to each other. B and C are similar, so det(A) = det(C)
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I. Answer "True" or "False" for each statement. If the statement is "true," provide justification (proof) your answer. If the statement is "false," provide a counter example that would be true. for i. If i is in both the nullspace and the column space of an n x n matrix A, then i = ỡ. ii. If A is a square matrix, NS(A) is the same as the eigenspace Eo · ii. If A is a square matrix, every vector is an eigenvector in the eigenspace E·
True or False

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