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)