We may define the determinant of order by induction as fallows:where Aij is the Cofacter of the element αij. From (1)

we may obtain directly all the properties of determinants

for example

Laplace development theorem and the rule of the multiplication of detarminants

etc. Finally

we hawe edentified (1) with the ordinary definition of the determinant of order n:where

the summation is over all permuttions p1 …… pn

of the set s={1

2

……n}

and [P1……Pn] is the inversion of the permutation p1……pn.