Let ** V** be a vector space. Let

`D` is multilinear if it is linear in each vector variable

`D`( ..., ` u` +

`D` is alternating if switching two variables introduces a negative sign

`D`( ..., ` u`, ...,

Note that taking ** u** =

0 = `D`( ..., ` u` +

=

=

=

which is the alternating property. Thus we proved the following.

`D`( ..., ` u`, ...,

A function `D`(** A**) of

First for a 2 by 2 matrix

[A = |
a |
b |
] = [ae_{1} + ce_{2}, be_{1} + de_{2}], |

c |
d |

the multilinearity and the alternating property give us

`D`(** A**) =

=

=

=

=

=

which is a constant `D`(`I`_{2}) multiplied to the determinant.

For the 3 by 3 case

= [A |
a_{11} |
a_{12} |
a_{13} |
] = [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_{13}e_{1} + a_{23}e_{2} + a_{33}e_{3}], |

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

a_{31} |
a_{32} |
a_{33} |

the multilinearity gives us

`D`(** A**) =

=

= ∑

where `i`, `j`, `k` can be any of 1, 2, 3. By the alternating property, if any two of `i`, `j`, `k` are the same, then `D`(` e_{i}`,

`D`(** A**) =

= ∑

Since `i`, `j`, `k` are distinct numbers chosen from 1, 2, 3, they must be a permutation (rearrangement of positions) of (1, 2, 3). A permutation (`i`, `j`, `k`) can be changed to (1, 2, 3) by a sequence of transpositions (exchange of *two* positions). For example, (2, 3, 1) is changed to (1, 2, 3) by

(2, 3, 1) → (2, 1, 3) → (1, 2, 3),

where the blue color highlights the transpositions. By the alternating property, the sequence of transpositions tells us

`D`(**e**_{2}, **e**_{3}, **e**_{1}) = - `D`(**e**_{2}, **e**_{1}, **e**_{3}) = `D`(**e**_{1}, **e**_{2}, **e**_{3}).

In general, we always have` D`(` e_{i}`,

permutation | transpositions to (1, 2, 3) | term in D() A |
equal to |

(1, 2, 3) | = (1, 2, 3) | a_{11}a_{22}a_{33}D(e_{1}, e_{2}, e_{3}) |
a_{11}a_{22}a_{33}D(e_{1}, e_{2}, e_{3}) |

(2, 3, 1) | → (2, 1, 3) → (1, 2, 3) | a_{21}a_{32}a_{13}D(e_{2}, e_{3}, e_{1}) |
a_{21}a_{32}a_{13}D(e_{1}, e_{2}, e_{3}) |

(3, 1, 2) | → (1, 3, 2) → (1, 2, 3) | a_{31}a_{12}a_{23}D(e_{3}, e_{1}, e_{2}) |
a_{31}a_{12}a_{23}D(e_{1}, e_{2}, e_{3}) |

(3, 2, 1) | → (1, 2, 3) | a_{31}a_{22}a_{13}D(e_{3}, e_{2}, e_{1}) |
-a_{31}a_{22}a_{13}D(e_{1}, e_{2}, e_{3}) |

(2, 1, 3) | → (1, 2, 3) | a_{21}a_{12}a_{33}D(e_{2}, e_{1}, e_{3}) |
-a_{21}a_{12}a_{33}D(e_{1}, e_{2}, e_{3}) |

(1, 3, 2) | → (1, 2, 3) | a_{11}a_{32}a_{23}D(e_{1}, e_{3}, e_{2}) |
-a_{11}a_{32}a_{23}D(e_{1}, e_{2}, e_{3}) |

Originally, `D`(** A**) is the sum of all the terms in the third column. By using the alternating property, this is equal to the sum of all the terms in the fourth column. Comparing with the definition of determinant for 3 by 3 matrices, we get

`D`(** A**) =

which is a constant `D`(`I`_{3}) multiplied to the determinant.

The pattern we saw in the computation of `D`(** A**) leads us to the following definition.

The determinant is a function `det` of `n` by `n` matrices, such that

`det`is multilinear in each column of**A**.`A``det`is alternating in columns of**A**.**A**`det`= 1.**I**_{n}

Several issues need to be settled in order to rigrously justify the definition.

- The function as described in the definition really exists;
- The function is also unique (so there is no ambiguity in the definition);
- The function has all the properties (see here, here, here) we gave to the determinant early on.