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Mathematics-Online course: Linear Algebra - Matrices - Determinants

Determinant as Antisymmetric Multilinear Form


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The determinant

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

of a square matrix $ A$ with columns $ a_j$ can be defined by the following properties: By the properties given above a determinant can be expanded into a sum of $ n$-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