# Determinant/Field/Recursively/Multiplication theorem/No proof/Section

We discuss without proofs further important theorems about the determinant. The proofs rely on a systematic account of the properties which are characteristic for the determinant, namely the properties multilinear and alternating. By these properties, together with the condition that the determinant of the identity matrix is , the determinant is already determined.

Let denote a field, and . Then for matrices , the relation

holds.

### Proof

The transposed matrix arises by interchanging the role of the rows and the columns. For example, we have

### Proof

This implies that we can compute the determinant also by expanding with respect to the rows, as the following statement shows.

For , the first equation is the recursive definition of the determinant. From that statement, the case follows, due to fact. By exchanging columns and rows, the statement follows in full generality, see exercise.