The determinant can be defined through multilinear and alternating properties. By extending the properties to nonsquare matrices, the concept of determinant can be extended.

For the 2 by 3 case

[A = |
a_{11} |
a_{12} |
a_{13} |
] = [a_{11}e_{1} + a_{21}e_{2}, a_{12}e_{1} + a_{22}e_{2}, a_{13}e_{1} + a_{23}e_{2}], |

a_{21} |
a_{22} |
a_{23} |

the multilinearity implies

`D`(** A**) =

=

=

+

=

+

+

+

Note that in each `D`(` e_{i}`,

Similarly, for the 3 by 2 case

= [A |
a_{11} |
a_{12} |
] = [a_{11}e_{1} + a_{21}e_{2} + a_{31}e_{3}, a_{12}e_{1} + a_{22}e_{2} + a_{32}e_{3}], |

a_{21} |
a_{22} |
||

a_{31} |
a_{32} |

the multilinearity implies

`D`(** A**) =

=

+

+

=

+

+

The alternating property further implies

`D`(** A**) = (

= D(e_{1}, e_{2}) det[ |
a_{11} |
a_{12} |
] + D(e_{1}, e_{3}) det[ |
a_{11} |
a_{12} |
] + D(e_{2}, e_{3}) det[ |
a_{21} |
a_{22} |
]. |

a_{21} |
a_{22} |
a_{31} |
a_{32} |
a_{31} |
a_{32} |

Thus the function `D`(** A**) is determined by three numbers

D() = AD_{12}det[ |
a_{11} |
a_{12} |
] + D_{13}det[ |
a_{11} |
a_{12} |
] + D_{23}det[ |
a_{21} |
a_{22} |
] |

a_{21} |
a_{22} |
a_{31} |
a_{32} |
a_{31} |
a_{32} |

is multilinear and alternating. A fancy way of saying this is that all the multilinear and alternating functions on 3 by 2 matrices form a vector space of dimension 3, with the three 2 by 2 determinants as a basis.

In general, for `m` by `n` matrix ** A**, the multilinearity gives us (see here)

`D`(** A**) = ∑

By the alternating property, `D`(**e**_{i1}, **e**_{i2}, ..., ` e_{in}`) ≠ 0 only if

Now assume `m` ≥ `n`. Then `D`(** A**) will be nontrivial. Moreover, we may rearrange each choice (

`D`(** A**) = ∑

Then by alternating property, we have

`D`(**e**_{i1}, **e**_{i2}, ..., ` e_{in}`) =

where `sign`(`i`_{1}, `i`_{2}, ..., `i _{n}`)

(6, 3, 8, 5) → (3, 6, 8, 5) → (3, 6, 5, 8) → (3, 5, 6, 8)

tells us `sign`(6, 3, 8, 5) = -1. Thus

`D`(** A**) = ∑

Now ∑_{all permutations (i1,i2,...,in) of J} `sign`(`i`_{1}, `i`_{2}, ..., `i _{n}`)

** A**(

then

∑_{all permutations (i1,i2,...,in) of J} `sign`(`i`_{1}, `i`_{2}, ..., `i _{n}`)

and we conclude the following.

Suppose `D`(** A**) is a multilinear and alternating function on columns of

`D`(** A**) = ∑

for `m` ≥ `n`.

Conversely, for arbitrarily chosen numbers `D _{J}` (one for each subset

Finally, the functions described in this section can be reformulated as linear functions over a suitable vector space Λ^{n}**R**^{m}, called exterior algebra, which is a fundamental tool in modern mathematics.

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