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Mathematics-Online course: Linear Algebra - Matrices - Determinants

Oriented Volume

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The absolute value of the determinant of a real matrix $ A$ coincides with the $ n$-dimensional volume of the parallelepiped spanned by the colums $ a_i$:

$\displaystyle \vert\operatorname{det}A\vert = \operatorname{vol}\{\sum_{i=1}^n \alpha_i a_i:\ 0\le\alpha_i\le 1\}\, . $

(Authors: Burkhardt/Höllig/Hörner)
It can easily be verified that the oriented volume

$\displaystyle \operatorname{vol}_*A := \operatorname{sign}(\operatorname{det} A) \operatorname{vol}(A[0,1]^n) $

has all three defining properties of the determinant (multilinearity, antisymmetry, normalisation). Thus, we obtain

$\displaystyle \operatorname{vol}_*A = \operatorname{det} A\,. $

(Authors: Burkhardt/Höllig/Hörner)

For vectors $ u$, $ v$, $ w$ in $ \mathbb{R}^3$ the determinant coincides with the parallelepidial product:

$\displaystyle \operatorname{det}(u,v,w) = [u,v,w] = \left\langle u,(v\times w)\right\rangle\, . $

Alternatively, the determinant can be expressed by means of the $ \varepsilon$-tensor:

$\displaystyle \operatorname{det}(u,v,w) = \sum_i \sum_j \sum_k \varepsilon_{i,j,k} u_iv_jw_k\, . $

Choosing, for example, the vectors

$\displaystyle u=(2,1,1)^{\operatorname t}\,,\quad v=(1,2,1)^{\operatorname t}\,,\quad w=(1,1,2)^{\operatorname t} $

the parallelepidial product yields

$\displaystyle \left\langle\left(\begin{array}{c}2\\ 1\\ 1\end{array}\right), \... ...left(\begin{array}{c}3\\ -1\\ -1\end{array}\right)\right\rangle\ =6-1-1=4\,. $

By means of the $ \varepsilon$-tensor we obtain the representation
$\displaystyle \operatorname{det}(u,v,w)$ $\displaystyle =$ $\displaystyle \varepsilon_{1,2,3}\cdot2\cdot2\cdot2 +\varepsilon_{1,3,2}\cdot2\cdot1\cdot1 +\varepsilon_{2,1,3}\cdot1\cdot1\cdot2$  
    $\displaystyle +\varepsilon_{2,3,1}\cdot1\cdot1\cdot1 +\varepsilon_{3,1,2}\cdot1\cdot1\cdot1 +\varepsilon_{3,2,1}\cdot1\cdot2\cdot1$  
  $\displaystyle =$ $\displaystyle 8-2-2+1+1-2=4\,.$  

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005