Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 11

namely

R

′

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

R

′

2 = R^2 +u^23 R^3

R

′

3 = R^3 ,

can be expressed in the form





R

′

1

R

′

2

R

′

3



=





1 u 12 u 13

1 u 23

1









R 1

R 2

R 3



.

Denote the upper triangular matrix byU 3. These operations, when per-

formed in the given order on an arbitrary determinantA 3 =|aij| 3 , have

the same effect aspremultiplication ofA 3 by the unit determinantU 3 .In

each case, the result is

A 3 =

∣ ∣ ∣ ∣ ∣ ∣ ∣

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 31 a 32 a 33

∣ ∣ ∣ ∣ ∣ ∣ ∣

.

(2.3.2)

Similarly, the column operations

C

′

i=

3

∑

j=i

uijCj,uii=1, 1 ≤i≤3; uij=0, i>j, (2.3.3)

when performed in the given order on A 3 , have the same effect as

postmultiplication ofA 3 byU
T
3

. In each case, the result is

A 3 =

∣ ∣ ∣ ∣ ∣ ∣ ∣

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

a 21 +u 12 a 22 +u 13 a 23 a 22 +u 23 a 23 a 23

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

∣ ∣ ∣ ∣ ∣ ∣ ∣

. (2.3.4)

The row operations

R

′

i=

i

∑

j=1

vijRj,vii=1, 1 ≤i≤3; vij=0,i<j, (2.3.5)

can be expressed in the form





R

′

1

R

′

2

R

′

3



=





1

v 21 1

v 31 v 32 1









R 1

R 2

R 3



.

Denote the lower triangular matrix byV 3. These operations, when per-

formedin reverse orderonA 3 , have the same effect aspremultiplication of

A 3 by the unit determinantV 3.