### Theory of Determinant

##### 3. Existence and uniqueness

Based on the three properties in the definition, we derived the formula

`det`**A** = ∑_{all permutations (i1,i2,...,in) of (1,2,...,n)} `sign`(`i`_{1}, `i`_{2}, ..., `i`_{n}) `a`_{i11}`a`_{i22}...`a`_{inn}.

We have two remarks.

- Since the functions satisfying the three properties must be given by the explicit formula, the determinant (if it exists) must be
*unique*;
- We may prove the
*existence* of `det` by verifying that the formula indeed satsify the three properties.

In the subsequent verification that the formula satisfies the three properties, the formula will be temporarily denoted `D`(`A`) (we are not supposed to use the `det` until the properties are verified).

Multilinear Since linear combinations of linear functions are linear, we only need to verify that for each fixed permutation (`i`_{1}, `i`_{2}, ..., `i`_{n}), the function `a`_{i11}`a`_{i22}...`a`_{inn} is linear in columns of `A`. For the linearity in [col 1], we fix entries in [col 2], ..., [col `n`] and consider entries in [col 1] as changing. In our case, `a`_{i22}...`a`_{inn} is a fixed number and `a`_{i11} changes. Then the composition

`a`_{1} = [ |
`a`_{11} |
] → `a`_{i11} → `a`_{i11}`a`_{i22}...`a`_{inn} |

`a`_{21} |

: |

`a`_{n1} |

is linear because the first projection is linear and the second scalar multiplication by a fixed number is also linear. The reason for the linearity in the other columns is similar.

Alternating Let `B` be obtained from `A` by exchanging [col 1] and [col 2]. Then `a`_{i2} is an entry in [col 1] of `B`, `a`_{i1} is an entry in [col 2] of `B`, and the others are not changed. The change leads to

`D`(**B**) = ∑_{all permutations (i1,i2,...,in) of (1,2,...,n)} `sign`(`i`_{1}, `i`_{2}, ..., `i`_{n}) `a`_{i12}`a`_{i21}`a`_{i33}...`a`_{inn}

= ∑_{all permutations (i1,i2,...,in) of (1,2,...,n)} `sign`(`i`_{1}, `i`_{2}, ..., `i`_{n}) `a`_{i21}`a`_{i12}`a`_{i33}...`a`_{inn}.

Now we claim

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

To see why, let

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

be a sequence of transpositions changing (`i`_{1}, `i`_{2}, ..., `i`_{n}) to (1, 2, ..., `n`). Then by adding the transposition (`i`_{2}, `i`_{1}, ..., `i`_{n}) → (`i`_{1}, `i`_{2}, ..., `i`_{n}) to the front, we get a sequence

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

of transpositions changing (`i`_{2}, `i`_{1}, ..., `i`_{n}) to (1, 2, ..., `n`). The upshort is that the number of transpositions in the two sequences have different parity. Thus `sign`(`i`_{2}, `i`_{1}, ..., `i`_{n}) and `sign`(`i`_{1}, `i`_{2}, ..., `i`_{n}) have different signs. With the help of the discussion on sign, we conclude (in the second equality, we relabeled `i`_{2} as `i`_{1} and `i`_{1} as `i`_{2})

`D`(**B**) = ∑_{all permutations} - `sign`(`i`_{2}, `i`_{1}, ..., `i`_{n}) `a`_{i21}`a`_{i12}`a`_{i33}...`a`_{inn}

= - ∑_{all permutations} `sign`(`i`_{1}, `i`_{2}, ..., `i`_{n}) `a`_{i11}`a`_{i22}...`a`_{inn}

= - `D`(**A**).

Identity In the identity matrix `I` we have `a`_{ij} = 0 for `i` ≠ `j` and `a`_{ii} = 1. Therefore the term `a`_{i11}`a`_{i22}...`a`_{inn} is nonzero only if the permutation (`i`_{1}, `i`_{2}, ..., `i`_{n}) = (1, 2, ..., `n`). Thus

`D`(`I`) = `sign`(1, 2, ..., `n`) 1×1×...×1 = 1.