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Mathematik-Online problems:

Problem 48: Calculation Rules for Determinants

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
Proof:
a)
For all $ a_1, a_2, b_1, b_2, c, d\in\mathbb{R}$ is true:

$\displaystyle \left\vert\begin{array}{cc} a_1+a_2 & b_1+b_2 \\ c & d \end{arra... ...ght\vert+\left\vert\begin{array}{cc} a_2 & b_2 \\ c & d \end{array}\right\vert $

b)
For all $ u, v, w \in\mathbb{R}^3$ and for all $ \alpha, \beta, \gamma\in\mathbb{R}$ is true:

$\displaystyle \left\vert\begin{array}{rrr} \alpha u_1 & \alpha v_1 & \alpha w_1... ...} u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3 \end{array}\right\vert $

c)
For all $ A\in\mathbb{C}^{n\times n}$ is true:          $ {\mathrm{det}}\big( \overline{A} \big)=\overline{{\mathrm{det}}\hspace*{0.05cm}(A)}$


Hint: Use the expansion theorem and induction.

(Authors: Kimmerle/Höfert)

see also:

[Solutions]
  automatisch erstellt am 29. 10. 2004