Page 26 - Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 11

namely

R = R 1 + u 12 R 2 + u 13 R 3
1

R =
2 R 2 + u 23 R 3
R = R 3 ,

3
can be expressed in the form
     
R 1
1 u 12 u 13 R 1
R  = 1 .
 2  u 23   R 2 
R 3 1 R 3
Denote the upper triangular matrix by U 3 . These operations, when per-
formed in the given order on an arbitrary determinant A 3 = |a ij | 3 , have
the same effect as premultiplication of A 3 by the unit determinant U 3 .In
each case, the result is

a 11 + u 12 a 21 + u 13 a 31 a 12 + u 12 a 22 + u 13 a 32 a 13 + u 12 a 23 + u 13 a 33

a 21 + u 23 a 31 a 22 + u 23 a 32 a 23 + u 23 a 33 .
A 3 =

a 31 a 32 a 33

(2.3.2)
Similarly, the column operations
3

C = u ij C j , u ii =1, 1 ≤ i ≤ 3; u ij =0, i>j, (2.3.3)

i
j=i
when performed in the given order on A 3 , have the same effect as
postmultiplication of A 3 by U . In each case, the result is
T
3

a 11 + u 12 a 12 + u 13 a 13 a 12 + u 23 a 13 a 13

A 3 = a 21 + u 12 a 22 + u 13 a 23 a 22 + u 23 a 23 a 23 . (2.3.4)

a 31 + u 12 a 32 + u 13 a 33 a 32 + u 23 a 33 a 33

The row operations
i

R = v ij R j , v ii =1, 1 ≤ i ≤ 3; v ij =0, i < j, (2.3.5)
i
j=1
can be expressed in the form
     
R 1 1 R 1
R  = 1 .
 2  v 21   R 2 
R 3 v 31 v 32 1 R 3
Denote the lower triangular matrix by V 3 . These operations, when per-
formed in reverse order on A 3 , have the same effect as premultiplication of
A 3 by the unit determinant V 3 .