There is also a combinatorial approach to the computation of the determinant.

Let be the set consisting of the integers between and inclusive. We denote by the set of maps of to itself which are isomorphisms, meaning that they are 1-1 and onto. An element should be thought of as a reordering of the elements of , with consisting of all such reorderings. Elements of can be multiplied (with multiplication given by composition), and with respect to that multiplication every element has an inverse satisfying . A set satisfying these properties is called a group, and is typically referred to as the permutation group on n letters.

Among the elements of are permutations of a particularly simple type, called transpositions. The permutation which switches two numbers and , while leaving all others fixed, will be labeled (note: is often written as , however this can be confused with the coordinate notation for points in , hence our choice not to use it).

It is not hard to see that any permutation can be written as a product of transpositions: The way of doing so is far from unique, but it turns out that - given - the number of transpositions used rewriting in this fashion is always even or always odd. This allows for the definition of the sign of a permutation:

With this concept established, the determinant may alternatively be defined as

Let A = , as in the last example of the previous section. Using the combinatorial definition above, compute . Verify that you get the same answer as before.
Let A = . Using the combinatorial definition above, compute as a sum of products of entries of . Verify that you get the same answer that you did by using the cofactor expansion definition in Exercise 1 of the previous section.