The determinant function can be defined by essentially two different methods. The advantage of the first definition—one which uses permutations—is that it provides an actual formula for det A, a fact of theoretical importance. The disadvantage is that, quite frankly, no one actually computes a determinant by this method.
Method 1 for defining the determinant. If n is a positive integer, then a permutation of the set S = {1, 2, …, n} is defined to be a bijective function—that is, a one-to-one correspondence—σ, from S to S. For example, let S = {1, 2, 3} and define a permutation σ of S as follows: