A function defined on square matrices. There are many equivalent definitions of the determinant, most of them useful. It is often useful to treat the determinant as a function of n vectors of dimension n. Here are a few:
 Geometrical
 The determinant of n vectors is the volume of the parallelopiped they span (this is only useful in ndimensional euclidian space R^{n}).
 Axiomatic
 The determinant is a (in fact, "the") function D of n vectors which is linear is each of them, satisfies D(...,a,...,b,...) = D(...,b,...,a,...) (for every interchange of 2 arguments) and D(e_{1},...,e_{n}) = 1 (for e_{1},...,e_{n} the standard basis vectors (1,0,...,0), ..., (0,...,0,1)).
 Explicit formula

The determinant of a matrix
a_{11} ... a_{1n}
a_{21} ... a_{2n}
.......
a_{n1} ... a_{nn}
is the sum over all even permutations p of the integers 1...n of the products of a_{i,p(i)}, minus the sum over all odd permutations q of the products of a_{i,q(i)}.
 Recursive formula

The determinant of the matrix above is the sum
∑_{i=1,...,n} (1)^{i+1} a_{1,i} D_{i},
where D_{i} is the determinant of the (n1)×(n1)matrix you get by deleting column i and row 1 of the matrix above.