# Determinant/Field/Recursively/Multilinearity/No proof/Section

We want to show that the recursively defined determinant is a "multilinear“ and "alternating“ mapping, where we identify

so a matrix is identified with the -tuple of the rows of the matrix. We consider a matrix as a tuple of columns

where the entries are row vectors of length .

Let be a field, and . Then the determinant

is multilinear. This means that for every , and for every choice of vectors , and for any , the identity

holds, and for , the identity

holds.

### Proof

Let be a field, and . Then the determinant

- If in two rows are identical, then . This means that the determinant is alternating.
- If we exchange two rows in , then the determinant changes with factor .

### Proof

Let be a field, and let denote an -matrix

over . Then the following statements are equivalent.- We have .
- The rows of are linearly independent.
- is invertible.
- We have .

The relation between rank, invertibility and linear independence was proven in fact. Suppose now that the rows are linearly dependent. After exchanging rows, we may assume that . Then, due to fact and fact, we get

Now suppose that the rows are linearly independent. Then, by exchanging of rows, scaling and addition of a row to another row, we can transform the matrix successively into the identity matrix. During these manipulations, the determinant is multiplied with some factor . Since the determinant of the identity matrix is , the determinant of the initial matrix is .