Page 20 - Determinants and Their Applications in Mathematical Physics

1.4 The Product of Two Determinants — 1 5

Comparing this result with (1.2.5),
n

|a ij | n = a ik A ik (1.3.14)
k=1
which is the expansion of |a ij | n by elements from row i and their cofactors.
From (1.3.1) and noting (1.3.5),
x 1 x 2 ··· x n =(y 1 + a 1j e j )(y 2 + a 2j e j ) ··· (y n + a nj e j )

= a 1j e j y 2 y 3 ··· y n + a 2j y 1 e j y 3 ··· y n
+ ··· + a nj y 1 y 2 ··· y n−1 e j
=(a 1j A 1j + a 2j A 2j + ··· + a nj A nj )e 1 e 2 ··· e n

n

= a kj A kj e 1 e 2 ··· e n . (1.3.15)
k=1
Comparing this relation with (1.2.5),
n

|a ij | n = a kj A kj (1.3.16)
k=1
which is the expansion of |a ij | n by elements from column j and their
cofactors.

1.4 The Product of Two Determinants — 1

Put
n

x i = a ik y k ,
k=1
n

y k = b kj e j .
j=1
Then,
x 1 x 2 ··· x n = |a ij | n y 1 y 2 ··· y n ,
y 1 y 2 ··· y n = |b ij | n e 1 e 2 ··· e n .
Hence,

x 1 x 2 ··· x n = |a ij | n |b ij | n e 1 e 2 ··· e n . (1.4.1)
But,

n n

x i = a ik b kj e j
k=1 j=1