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:

	* for n=1, det(M) = m(1,1).
	* for n>1, det(M) = sum from (k = 1 to n) of m(x, k) * det(M(x, k))

where M(i,j) is the (n-1)-by-(n-1) matrix formed by deleting row i and column j of M, and x may be chosen as any value from 1 to n (but is constant over the summation).

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. 
	* The usual approach is to handwave away the meaning of probability, appealing to intuition, to that "nondeterminism" gibberish that is so common in QuantumPhysics. 
	* Some old books give the formal definition in terms of mappings and such.
	* HowToLieWithStatistics shows, well, how to lie with statistics.  This teaches the basics in a very fun, practical way.

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RefactorMe:  This page is wandering from the topic of MatrixDeterminant''''''s, into opinions about the teaching of mathematics.

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CategoryMath