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Mathematik-Online problems:

Problem 54: States on Eigenvalues of Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the $ n\times n$-matrices $ A$ and $ B$. Let $ \lambda$ be an eigenvalue of $ A$ and let $ \mu$ be an eigenvalue of $ B$. Mark which states are true respectively false and give reasons for your answers.

$ A$ is not a regular matrix $ \Longleftrightarrow$ all eigenvalues of $ A$ are 0  true $ \bigcirc $  false $ \bigcirc $
$ A$ is a regular matrix $ \Longleftrightarrow$ all eigenvalues of $ A$ are $ \neq 0$  true $ \bigcirc $  false $ \bigcirc $
$ \lambda^n$ is always eigenvalue of $ A^n$  true $ \bigcirc $  false $ \bigcirc $
$ \lambda\mu$ is always eigenvalue of $ AB$  true $ \bigcirc $  false $ \bigcirc $
$ \lambda+\mu$ is always eigenvalue of $ A+B$  true $ \bigcirc $  false $ \bigcirc $
$ A$ is a regular matrix $ \Longrightarrow$ $ \lambda^{-1}$ is eigenvalue of $ A^{-1}$  true $ \bigcirc $  false $ \bigcirc $
$ A^m=0_n$, for a $ m\in\mathbb{N}$ $ \Longleftrightarrow$ 0 is eigenvalue of $ A$  true $ \bigcirc $  false $ \bigcirc $

(Authors: Blind/Simon/Kimmerle/Höfert)

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[Solutions]
  automatisch erstellt am 14. 10. 2004