Suppose we have an n-by-n matrix M, with m(i,j) being the entry of M in the i'th row and j'th column. The determinant is found recursively as follows:
This is useful for finding the inverse of a matrix (MatrixInverse, which has an example for a 2x2 matrix); see CofactorMatrix for more information.
-- KarlKnechtel
There's already a pretty extensive math web online.
Also, the recursive definition of the determinant of a matrix is useful to compute it, but not to understand its transcendent meaning. For that purpose, it's better to define it as the sum of the product of every permutation of rows and columns, with each product multiplied by -1 if its corresponding permutation is odd. Naturally, only old books will give you this latter definition.
The difference between computation and comprehension is vast. Many academic fields, math and physics being prime examples, are taught with little regard to comprehension. For instance, all of calculus with its delta-epsilon proofs is vastly less comprehensible than non-standard math. Yet a delta-epsilon approach is still used to teach calculus.
The same is true of the mathematical concept of probability.
RefactorMe: This page is wandering from the topic of MatrixDeterminants, into opinions about the teaching of mathematics.