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Exercise 4: (1) Is it possible to find a 2 × 2 matrix A such that A 0 but A² = 0? If yes find one, if not explain why not. (2) Given a square matrix A, is it always true that Null(A) is contained in Null(A2)? Explain your answer. (3) Given a square matrix A, is it always true that dim (Col(A)) < dim(Col(A²))? Explain your answer.

Exercise 4: (1) Is it possible to find a 2 × 2 matrix A such that A 0 but A² = 0? If yes find one, if not explain why not. (2) Given a square matrix A, is it always true that Null(A) is contained in Null(A2)? Explain your answer. (3) Given a square matrix A, is it always true that dim (Col(A)) < dim(Col(A²))? Explain your answer.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section: Chapter Questions
Problem 17RQ
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Exercise 4: (1) Is it possible to find a 2 × 2 matrix A such that A 0 but A² = 0? If yes find one, if not explain why not. (2) Given a square matrix A, is it always true that Null(A) is contained in Null(A²)? Explain your answer. (3) Given a square matrix A, is it always true that dim (Col(A)) < dim(Col(A²))? Explain your answer.
Transcribed Image Text:Exercise 4: (1) Is it possible to find a 2 × 2 matrix A such that A 0 but A² = 0? If yes find one, if not explain why not. (2) Given a square matrix A, is it always true that Null(A) is contained in Null(A²)? Explain your answer. (3) Given a square matrix A, is it always true that dim (Col(A)) < dim(Col(A²))? Explain your answer.
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