Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 13

The elements come from rowiofA, but the cofactors belong to the elements

in rowkand are said to be alien to the elements. The identity is merely

an expansion by elements from rowkof the determinant in which rowk=

rowiand which is therefore zero.

The identity can be combined with the expansion formula forAwith the

aid of the Kronecker delta functionδik(Appendix A.1) to form a single

identity which may be called the sum formula for elements and cofactors:

n

∑

j=1

aijAkj=δikA, 1 ≤i≤n, 1 ≤k≤n. (2.3.12)

It follows that

n

∑

j=1

AijCj=[0... 0 A 0 ...0]

T

, 1 ≤i≤n,

where the elementAis in rowiof the column vector and all the other

elements are zero. IfA= 0, then

n

∑

j=1

AijCj=0, 1 ≤i≤n, (2.3.13)

that is, the columns are linearly dependent. Conversely, if the columns are

linearly dependent, thenA=0.

2.3.5 Cramer’s Formula

The set of equations

n

∑

j=1

aijxj=bi, 1 ≤i≤n,

can be expressed in column vector notation as follows:

n

∑

j=1

Cjxj=B,

where

B=

[

b 1 b 2 b 3 ···bn

]T

.

IfA=|aij|n= 0, then the unique solution of the equations can also be

expressed in column vector notation. Let

A=

∣

∣C

1 C 2 ···Cj···Cn

∣

∣.

Then

xj=

1

A

∣

∣C

1 C 2 ···Cj− 1 BCj+1···Cn

∣

∣