and the expanded value is

Each term in (5) is of the form with the plus sign used if i, j, k is an even permutation of 1, 2, 3 and the minus sign if i, j, k is an odd permutation. The terms in (5) can be grouped in several ways, one of which is

The coefficient a₂₁a₃₂ - a₂₂a₃₁ of a₁₃ in (6) is the value of the 2 by 2 determinant that results when the entire first row and third column of D are removed; the coefficients of a₁₁ and -a₁₂ in (6) can be characterized similarly.

This motivates the following definitions: Let a_ij be a given entry in the determinant D of (4) and let M_ij be the 2 by 2 determinant obtained by deleting the ith row and the jth column of D. This determinant M_ij is called the minor of the entry a_ij. The cofactor C_ij of the entry a_ij is defined by the formula

For example, the minor of a₂₃ is

and the cofactor of a₂₃ is

It now can easily be seen that equation (6) may be rewritten as
 

or as

Wednesday, June 10, 1998