Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 33

Exercises

1.Ifn= 4, prove that

∑

p<q

N 23 ,pqA 24 ,pq=

∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 a 13 a 14

a 21 a 22 a 23 a 24

a 31 a 32 a 33 a 34

a 31 a 32 a 33 a 34

∣ ∣ ∣ ∣ ∣ ∣ ∣

=0

(row4=row3),byexpanding the determinant from rows 2 and 3.

2.Generalize the sum formula for the caser=3.

3.3.5 The Product of Two Determinants — 2

Let

An=|aij|n

Bn=|bij|n.

Then

AnBn=|cij|n,

where

cij=

n

∑

k=1

aikbkj.

A similar formula is valid for the product of two matrices. A proof has

already been given by a Grassmann method in Section 1.4. The following

proof applies the Laplace expansion formula and row operations but is

independent of Grassmann algebra.

Applying in reverse a Laplace expansion of the type which appears in

Section 3.3.3,

AnBn=

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 ... a 1 n

a 21 a 22 ... a 2 n

... ... ... ...

an 1 an 2 ... ann

− 1 b 11 b 12 ... b 1 n

− 1 b 21 b 22 ... b 2 n

... ... ... ... ...

− 1 bn 1 bn 2 ... bnn

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 n

. (3.3.15)

Reduce all the elements in the firstnrows and the firstncolumns, at

present occupied by theaij, to zero by means of the row operations

R

′

i=Ri+

n

∑

j=1

aijRn+j, 1 ≤i≤n. (3.3.16)