Mathematics-Online course: Linear Algebra-Matrices-Determinants-Determinant as Antisymmetric Multilinear Form

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	Mathematics-Online course: Linear Algebra - Matrices - Determinants
	Determinant as Antisymmetric Multilinear Form

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[table of contents][page overview]

The determinant

$\displaystyle \operatorname{det} A = \operatorname{det}(a_1,\ldots,a_n)$

of a square matrix

with columns

can be defined by the following properties:

Multilinearity:

$\displaystyle \operatorname{det}(\ldots,\alpha a_j+\beta b_j,\ldots) = \alph... ...{det}(\ldots,a_j,\ldots) + \beta \operatorname{det}(\ldots,b_j,\ldots)\, .$
Antisymmetry:

$\displaystyle \operatorname{det}(\ldots,a_j,\ldots,a_k,\ldots) = -\operatorname{det}(\ldots,a_k,\ldots,a_j,\ldots)\, .$
Scaling:

$\displaystyle \operatorname{det}(e_1,\ldots,e_n) = 1\, .$

By the properties given above a determinant can be expanded into a sum of

-fold products:

$\displaystyle \operatorname{det}A = \sum_{i \in S_n} \sigma(i) a_{i_1,1}\cdots a_{i_n,n}\, ,$

where summation goes over all permutations $(i_1,\ldots,i_n)$ of $(1,\ldots,n)$ and $\sigma$ denotes the sign of the permutation.

The following notation is also used:

$\displaystyle \operatorname{det} A = \vert A\vert = \left\vert\begin{array}... ...vdots & & \vdots \\ a_{n,1} & \cdots & a_{n,n} \end{array}\right\vert\, .$

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automatically generated 4/21/2005