. ,. .
141
I:. J.
Vo. [1978) 141-14
CONCERNING ROW OPERATIONS ON SYSTEMS
OF LINEAR EQUATIONS WITH INTEGRAL COEFFICIENTS
L. B. TREYBIG
Department o f Mathematics
Texas A & M University
College Station, Texas 77843
(Received January 20, 1978)
While trying to solve the problem stated in the introduction of [i] the author
Let
became aware of the theorem contained in this paper.
allXl +
+ alnXn
b
amxI +
+
b
amnXn
1
m
be a system of linear equations, where each
aij and b i is an integer.
n
b is replaced by
Define a row operation on (i) to be where
JffilZ aljxj
n
Z
(aij
+
b
takj)Xj
THEOREM i.
+
i
tbk,
where k
i
i and t is an integer.
Such a system as (i) can be transformed by a finite sequence of
row operations into
n
where (i)
(3)
dll
d2 j=IZ
<
PROOF
define E(1)
E(1) > E(2) >
d21
or
+
cx
+
(2)
c2j
d
CllX1 +
2
d
j=Z1 Icjl.
>
d
CmnXn
dm,
1
(2)
I +
n
i
j=IZ
cij
0
if
m_>
i > 2, and
O.
Given such a system as
m
ClnXn
(1), define c
n
7.
-bj(l
< j <
m), and
p=l aj P
Suppose we consider systems (i), (2),...,(q) where
3
E(q), but no row operation can be performed on (q) to find a
142
L.B. TREYBIG
For convenience suppose the coefficients of (q) are renamed as those
smaller E.
in (i).
Suppose there exist two integers
Thus
cj i
>
cj 2
cj i
n
E
j=l
sum.
aj I j
b.
31
by
n
J _ E1
or
(ajlJ
a
in
cj I + cj 2 I"
>
cj i
Jl’ J2
Icj 1
[1,m] such that
>
Icj 2
In the first case replace
J2j )xj bj i bj2
and in the other case use the
In either case the resulting system has a smaller E, so in all but one of the
equations of (q) the righthand coefficient is the sum of the other coefficients.
Using row operations we may suppose row one is the exception.
Still assuming the coefficients of (q) are the
n
D.
3
E a. + b. and let N(q)
3
p=l 3P
m
j 1
..IDjl-
aij
and b
i
define
We now suppose N has been reduced as
far as posslble by row operations which do not involve row one.
all but one j in
{2...,m},
Since
Dj
0 for
we suppose the exceptlonal row is two, and that our
system is in the form of (2) where conclusions (i) and (2) hold.
To show (3), if d 2 # 0 and
dll
>_
d21
we may subtract some integer multiple
of row two from row one to produce the desired result.
REFERENCES
i.
Borosh, I. and L. B. Treybig. Bounds on Positive Integral Solutions of Linear
Diophantine Equations, Proc. Amer. Math. Soc. 55 (1976) 299-304.
KEY WORDS AND PHRASES.
Linear
eqns.
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
15A06, IOB05.