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 be a
field,
and let
be an
-matrix
over . For
,
let be the matrix which arises from , by leaving out the -th row and the -th column. Then
(for
and for every fixed and )
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.