Number Theory: An Introduction to Mathematics

228 V Hadamard’s Determinant Problem

(ii)if the matrix B is obtained from the matrix A by multiplying all entries in one row
byλ,thendetB=λdetA.

Proof This is also clear, since in the expression for detAeach summand contains
exactly one factor from any given row.

(iii)if two rows of A are the same,thendetA=0.

Proof Suppose for definiteness that the first and second rows are the same, and letτ
be the permutation which interchanges 1 and 2 and leaves fixed everyk>2. Thenτ
is odd and we can write

detA=

∑

σ∈An

α 1 σ 1 α 2 σ 2 ···αnσn−

∑

σ∈An

α 1 στ 1 α 2 στ 2 ···αnστn,

whereAnis the alternating group of all even permutations. In the second sum

α 1 στ 1 α 2 στ 2 ···αnστn=α 1 σ 2 α 2 σ 1 α 3 σ 3 ···αnσn=α 2 σ 2 α 1 σ 1 α 3 σ 3 ···αnσn,

because the first and second rows are the same. Hence the two sums cancel.

(iv)if the matrix B is obtained from the matrix A by adding a scalar multiple of one
row to a different row,thendetB=detA.

Proof Suppose for definiteness thatBis obtained fromAby addingλtimes the sec-
ond row to the first. Then

detB=

∑

σ∈Sn

(sgnσ)α 1 σ 1 α 2 σ 2 ···αnσn+λ

∑

σ∈Sn

(sgnσ)α 2 σ 1 α 2 σ 2 ···αnσn.

The first sum is detAand the second sum is 0, by (iii), since it is the determinant of
the matrix obtained fromAby replacing the first row by the second.

(v)if A is singular,thendetA=0.

Proof IfAis singular, then some row ofAis a linear combination of the remaining
rows. Thus by subtracting from this row scalar multiples of the remaining rows we can
replace it by a row of 0’s. For the new matrixBwe have detB=0, by (i). On the
other hand, detB=detA,by(iv).

(vi)if A=diag[δ 1 ,...,δn],thendetA=δ 1 ···δn.In particular,detDα=α.

Proof In the expression for detAthe only possible nonzero summand is that for which
σis the identity permutation, and the identity permutation is even.

(vii) det(AB)=detA·detB for all A,B∈Mn.

Proof IfAis singular, thenABis also and so, by (v), det(AB)= 0 =detA·detB.
Thus we now suppose thatAis nonsingular. Then there existsS ∈ SLn(F)such
thatSA=Dδfor some nonzeroδ ∈ F. Since, by the definition ofSLn(F),left
multiplication by S corresponds to a finite number of operations of the type
considered in (iv) we have