Determinants and Their Applications in Mathematical Physics

1.4 The Product of Two Determinants — 1 5

Comparing this result with (1.2.5),

|aij|n=

n

∑

k=

aikAik (1.3.14)

which is the expansion of|aij|nby elements from rowiand their cofactors.

From (1.3.1) and noting (1.3.5),

x 1 x 2 ···xn=(y 1 +a 1 jej)(y 2 +a 2 jej)···(yn+anjej)

=a 1 jejy 2 y 3 ···yn+a 2 jy 1 ejy 3 ···yn

+···+anjy 1 y 2 ···yn− 1 ej

=(a 1 jA 1 j+a 2 jA 2 j+···+anjAnj)e 1 e 2 ···en

=

[

n

∑

k=

akjAkj

]

e 1 e 2 ···en. (1.3.15)

Comparing this relation with (1.2.5),

|aij|n=

n

∑

k=

akjAkj (1.3.16)

which is the expansion of |aij|nby elements from columnj and their

cofactors.

1.4 The Product of Two Determinants — 1...........

Put

xi=

n

∑

k=

aikyk,

yk=

n

∑

j=

bkjej.

Then,

x 1 x 2 ···xn=|aij|ny 1 y 2 ···yn,

y 1 y 2 ···yn=|bij|ne 1 e 2 ···en.

Hence,

x 1 x 2 ···xn=|aij|n|bij|ne 1 e 2 ···en. (1.4.1)

But,

xi=

n

∑

k=

aik

n

∑

j=

bkjej