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Mathematik-Online problems:

Problem 83: True-False: Matrices et al.

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

a)
Let $ n\in\mathbb{N}$, $ A\in\mathbb{R}^{n\times n}$ and $ B=$ $ \left(\begin{array}{cc} 0 & a \\ b & 0 \end{array}\right)$, with $ a, b>0$.
Mark which of the following statements are always true respectively false, and give reasons for your answers.

$ 4^n-1$ is divisible by 3  true $ \bigcirc $  false $ \bigcirc $
$ A$ orthogonal $ \Longleftrightarrow \ {\mathrm{det}}\,(A)=\pm 1$  true $ \bigcirc $  false $ \bigcirc $
$ A$ and $ A^{\rm {t}}$ have the same eigenvalues  true $ \bigcirc $  false $ \bigcirc $
$ B$ has the eigenvalues $ a$ and $ b$  true $ \bigcirc $  false $ \bigcirc $
b)
Let $ x, y, z$ be vectors in $ \mathbb{R}^2$ of similar lengths. Each two of them enclose an angle of $ 120^\circ$. Give $ z$ as linear combination of $ x$ and $ y$.

(Authors: App/Apprich/Blind/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004