1 What is a Determinant? 225theory. The diffusion of this theory throughout the mathematical world owes much to
the clear exposition of Jacobi (1841).
For the practical solution of linear equations Cramer’s rule is certainly inferior to
the age-old method of elimination of variables. Even many of the theoretical uses to
which determinants were once put havebeen replaced by simpler arguments from
linear algebra, to the extent that some have advocated banning determinants from
the curriculum. However, determinants have a geometrical interpretation which makes
their survival desirable.
LetMn(R)denote the set of alln×nmatrices with entries from the real fieldR.
IfA∈Mn(R), then the linear mapx→AxofRninto itself multiplies the volume of
any parallelotope by a fixed factorμ(A)≥0. Evidently
(i)′′μ(AB)=μ(A)μ(B)for allA,B∈Mn(R),
(ii)′′μ(D)=|α|for any diagonal matrixD=diag[1,..., 1 ,α]∈Mn(R).
(A matrixA=(αjk)is denoted by diag[α 11 ,α 22 ,...,αnn]ifαjk=0 wheneverj=k
and is then said to bediagonal.) It may be shown (e.g., by representingAas a product
of elementary matrices in the manner described below) thatμ(A)=|detA|. The sign
of the determinant also has a geometrical interpretation: detA≷0 according as the
linear mapx→Axpreserves or reverses orientation.
Now letFbe an arbitrary field and letMn=Mn(F)denote the set of alln×n
matrices with entries fromF. We intend to show that determinants, as defined above,
have the properties:
(i)′det(AB)=detA·detBfor allA,B∈Mn,
(ii)′detD=αfor any diagonal matrixD=diag[1,..., 1 ,α]∈Mn,
and, moreover, that these two properties actually characterize determinants. To avoid
notational complexity, we consider first the casen=2.
LetEdenote the set of all matricesA∈M 2 which are products of finitely many
matrices of the formUλ,Vμ,where
Uλ=[
1 λ
01]
, Vμ=[
10
μ 1]
,
andλ,μ∈F.ThesetEis a group under matrix multiplication, since multiplication
is associative,I ∈E,Eis obviously closed under multiplication andUλ,Vμhave
inversesU−λ,V−μrespectively.
We are going to show that,if A∈M 2 and A=O,then there exist S,T∈Eand
δ∈F such that S AT=diag[1,δ].
For anyρ=0, put
W=
[
0 − 1
10
]
, Rρ=[
ρ−^10
0 ρ]
.
ThenW=U− 1 V 1 U− 1 ∈Eand alsoRρ∈Esince, ifσ = 1 −ρ,ρ′=ρ−^1 and
τ=ρ^2 −ρ,then
Rρ=V− 1 UσVρ′Uτ.