Section 5.2 Problems with Permutations and Cofactors

The determinant of an n by n matrix A can be expressed in more than one way:

The pivot formula comes mainly from the PA = LU theorem and property 9 of determinants:

The pivot formula is by far the most convenient to compute in the case n is large.

The permuation formula comes mainly from the linearity property of determinants and expresses it is as a sum of the n! terms:

det(A) = ,

where runs over the permuations of 1 to n and is the permutation matrix determined by . Any of the terms in which an occurs drops out. The permuation formula is an important theoretical tool in the study of linear algebra.

The cofactor formula also derives principally from the row linearity property 3 of determinants: for each row i of A,

= = ,

where is the cofactor defined by and is the (n-1) by (n-1) matrix obtained by removing the ith row and jth column from A. This formula is useful when working with matrices which have a fair number of 0's or are patterned in some nice way.