Advanced Calculus fi..

10 Advanced Calculus. Fifth EditionThen higher-order determinants are reduced to those of lower order. For example,From these formulas, one sees that a determinant of order n is a sum of terms, eachof which is f 1 times a product of n factors, one each from the n columns of thearray and one each from the n rows of the array. Thus from (1.3 1) and (1.30), oneobtains the six termsWe now state six rules for determinants:I. Rows and columns can be interchanged. For example,1;: ; ;1=1!; :;I.61 c~ a2 a3Hence in every rule the words row and column can be interchanged.11. Interchanging two rows (or columns) multiplies the determinant by -1. Forexample,Ill. A factor of any row (or column) can be placed before the determinant. Forexample,IV lf two rows (or columns) are proportional, the determinant equals 0. Forexample,1 1kal kbl kcla bl c~ = 0.a2 b2 C2V Determinants differing in only one row (or column) can be added by addingcorresponding elements in that row and leaving the other elements unchanged.For example,