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Let A be an n by n matrix. (a) Show that Nul(A - I) C Col(A). (b) If A² = A holds. Prove that Col(A) = Nul(A - I). Let A and B be n by n matrices. If A is invertible and AB is diagonalizable, prove that BA is also diagonalizable.

Let A be an n by n matrix. (a) Show that Nul(A - I) C Col(A). (b) If A² = A holds. Prove that Col(A) = Nul(A - I). Let A and B be n by n matrices. If A is invertible and AB is diagonalizable, prove that BA is also diagonalizable.

Advanced Engineering Mathematics
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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### Linear Algebra Problems #### Problem 1 Let \( A \) be an \( n \) by \( n \) matrix. **(a)** Show that \( \text{Nul}(A - I) \subseteq \text{Col}(A) \). **(b)** If \( A^2 = A \) holds, prove that \( \text{Col}(A) = \text{Nul}(A - I) \). #### Problem 2 Let \( A \) and \( B \) be \( n \) by \( n \) matrices. If \( A \) is invertible and \( AB \) is diagonalizable, prove that \( BA \) is also diagonalizable.
Transcribed Image Text:### Linear Algebra Problems #### Problem 1 Let \( A \) be an \( n \) by \( n \) matrix. **(a)** Show that \( \text{Nul}(A - I) \subseteq \text{Col}(A) \). **(b)** If \( A^2 = A \) holds, prove that \( \text{Col}(A) = \text{Nul}(A - I) \). #### Problem 2 Let \( A \) and \( B \) be \( n \) by \( n \) matrices. If \( A \) is invertible and \( AB \) is diagonalizable, prove that \( BA \) is also diagonalizable.
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