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Mathematik-Online problems:

Problem 23: Linear Independence of Matrices, Matrix Powers

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 matrices

$\displaystyle A=\left( \begin{array}{rrr} 2 & -6 & 6 \\ 3 & -7 & 6 \\ 3 & -6 ... ...\begin{array}{lll} 1 & x & x^2 \\ 0 & 1 & x \\ 0 & 0 & 1 \end{array}\right). $

a)
Calculate $ A^2$ and test, if the unitiy matrix $ E$ and the matrices $ A$ and $ A^2$ are linearly independent in the vector space of the $ 3\times 3$-matrices over the reals.
b)
Calculate $ B^2$ and $ B^3$ and use induction to derive a formula for $ B^{\mathit n}$, $ n\in\mathbb{N}$.

(Authors: Walk/Werner/Höfert)

see also:

[Solutions]
  automatisch erstellt am 18. 10. 2004