given above, the three-by-three determinant D consists of a sum of products of the form where i, j, k is a permutation of 1, 2, 3, and the plus sign is chosen when the permutation is even and the minus sign when it is odd. (For a definition of even and odd permutations, see Chapter 7.) Since there is a term corresponding to each permutation, the number of terms is 3! = 6, half preceded by a plus sign and half by a minus sign. (See Problem 21, Chapter 7.) Is should be noted that these observations also apply to two-by-two determinants

in that here the permutation 1, 2 is even and the permutation 2, 1 is odd, so that the 2! = 2 terms are preceded by the appropriate signs.

Example. Evaluate the determinant of the following system and thus show that the system has a unique solution:

Since  there is a unique solution.
 

Tuesday, June 23, 1998