ALGEBRA 13 



#11 din O O 



#ni a nn o o 



O O 11 bin 



O O b n i b nn 



DIFFERENTIATION OF DETERMINANTS 



1.330 If the elements of a determinant, A, are functions of a variable, t: 

 #A _ #'n 012 a\ n + an #'12 din 



dt d 21 #22 - #2n (hi (I 22 #2n 



a' ni #n2 a nn #ni #'n2 ...... a nn 



+ + #11 #12 a' In 



#21 #22 # 2n 



#nl #n2 # nn 



where the accents denote differentiation by t. 



EXPANSION OF DETERMINANTS 



1.340 The complete expansion of a determinant of the nth order contains n\ 

 terms. Each of these terms contains one element from each row and one ele- 

 ment from each column. . Any term may be obtained from the leading term : 



#11#22#33 ........ #nn 



by keeping the first suffixes unchanged and permuting the second suffixes among 

 i, 2, 3, . . . ., n. The sign of any term is determined by the number of inversions 

 from the second suffixes of the leading term, being positive if there is an even 

 number of inversions and negative if there is an odd number of inversions. 



1.341 The coefficient of an when the determinant A is fully expanded is: 



