Page 48 - Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 33

Exercises
1. If n = 4, prove that

a 11 a 12 a 13 a 14

a 21 a 22 a 23 a 24

=0
N 23,pq A 24,pq =
a 31 a 32 a 33 a 34
p<q

a 31 a 32 a 33 a 34
(row4=row3),by expanding the determinant from rows 2 and 3.
2. Generalize the sum formula for the case r =3.
3.3.5 The Product of Two Determinants — 2
Let

A n = |a ij | n
B n = |b ij | n .
Then
A n B n = |c ij | n ,

where
n

c ij = a ik b kj .
k=1
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,
...
a 11 a 12 a 1n
...
a 21 a 22 a 2n
... ... ... ...

a n1 a n2 ... a nn . (3.3.15)
−1 b 11 b 12 ... b 1n
A n B n =
−1 ...

b 21 b 22 b 2n
... ... ... ...

...
−1 b n1 b n2 ... b nn 2n

Reduce all the elements in the first n rows and the first n columns, at
present occupied by the a ij , to zero by means of the row operations
n

R = R i + a ij R n+j , 1 ≤ i ≤ n. (3.3.16)
i
j=1