5.4 Definition of the Determinant

We need some background knowledge before we can discuss the definition of the determinant. We want to form a product by choosing n elements where A is an n by n matrix. There will only be one element from each row and one element from each column in this product. For example, if one element of the product is a₂₁, then no other element in this product will be from row 2 or column 1. Let's look at a 3 by 3 matrix for an example.

We know that we will use an element from each column, so, for consistency, we will order the product this way: a_{_1}a_{_2}a_{_3}. We can fill in the blanks with row numbers. If we choose to begin with a₃₁, then we can choose from rows 1 and 2 for the remaining positions. One possible product formed by these rules is a₃₁a₁₂a₂₃. Another possible product is a₁₁a₃₂a₂₃. There are 3!, or 3*2*1 = 6, of these products. For our n by n matrix, there are n! possible products. All 6 possible products for this 3 by 3 matrix are: a₁₁a₂₂a₃₃, a₂₁a₃₂a₁₃, a₃₁a₁₂a₂₃, a₃₁a₂₂a₁₃, a₂₁a₁₂a₃₃, and a₁₁a₃₂a₂₃. Now, we need to determine which sign ( + or -) should be attached to each product. To do this, you need to order the product with the column numbers increasing as we did above and look at the sequences of row numbers. For the product, a₁₁a₂₂a₃₃, we look at the row sequence (1,2,3). We are looking for inversions, or numbers that are out of order. Since 1 comes before 2, 1 comes before 3, and 2 comes before 3 in the sequence, there are no inversions in this sequence. In the sequence (2,3,1), which comes from the product a₂₁a₃₂a₁₃, there are two inversions because 2 is placed before 1 and 3 is placed before 1. There are also two inversions for the sequence from the product a₃₁a₁₂a₂₃. There are three inversions for (3,2,1) and one inversion each for (2,1,3) and (1,3,2). If the number of inversions is even, then the sign attached to the product is positive. If the number of inversions is odd, then the sign attached to the product is negative. Notice that we did not say that the product was positive or negative. We simply are determining whether the product will be added or subtracted.

Definition 5.1 The determinant of a square matrix is the sum of all the n possible signed products formed from the matrix using each row and each column only once for each product. The sign to be attached to the product is the same as the one determined by the formula (-1)^N when N is the number of inversions as described above.