The Reduced Row Echelon Form of a Matrix is Unique: A Simple Proof
Author(s): Thomas Yuster
Source: Mathematics Magazine, Vol. 57, No. 2 (Mar., 1984), pp. 93-94
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2689590 .
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The Reduced Row Echelon Form of a
MatrixIs Unique: A Simple Proof
THOMAS
YUSTER
Middlebury
College
VT 05753
Middlebury,
forsolvingsystems
of linearequationsis to
One of themostsimpleand successful
techniques
by
matrixof thesystemto reducedrowechelonform.Thisis accomplished
reducethecoefficient
rowoperations
matrixuntil
(see,e.g.,[1,p. 5]) to thecoefficient
applyinga sequenceofelementary
thefollowing
a matrixB is obtainedwhichsatisfies
description:
If a rowofB does notconsistentirely
ofzerosthenthefirstnonzeronumberin
therowis a 1 (usuallycalleda leading1).
at
If thereare anyrowsthatconsistentirely
ofzeros,theyare groupedtogether
thebottomof B.
In anytwosuccessive
non-zerorowsof B, theleading1 in thelowerrowoccurs
farther
to therightthantheleading1 in thehigherrow.
else.
Each columnof B thatcontainsa leading1 has zeroseverywhere
The matrixB is said to be in reducedrowechelonform.
Ax = 0
It is wellknownthatifA is an m X n matrixand x is an n X 1 vector,thenthesystems
and Bx = 0 have thesamesolutionset.However,thesolutionto thesystemBx = 0 maybe read
fromthematrixB. For example,considerthesystem:
offimmediately
x1 + 2X2+ 4X3= 0
2x1 + 3X2+ 7X3= 0
3x1+ 3X2+ 9X3= 0
is
The matrixof coefficients
1 24
2 3
3 3 9
and through
use of elementary
we obtain
rowoperations
A=
B=
0
1
whichis seen to be in reducedrowechelonform.Column3 (unlikecolumns1 and 2) does not
containa leading1. We call sucha columna freecolumn,becausein thesolutionto thesystem
Thus
to thatcolumnis a freeparameter.
Bx = 0 (and henceAx = 0), thevariablecorresponding
we can set X3= t, and read offthe equationsxl + 2t = 0 (row 1) and X2+ t = 0 (row 2). The
solutionsetis thenxl = - 2t, X2 = - t, X3 = t.
resultis thatthereducedrowechelonformof a matrixis unique.
An important
theoretical
or givea proofwhichis longand verytechnical(see [2,
Most textseitheromitthisresultentirely
p. 56]). The following
proofis somewhatclearerand less complicatedthanthestandardproofs.
THEOREM.
Thereducedrowechelon
formofa matrixis unique.
Proof.Let A be an mX n matrix.We willproceedby inductionon n. For n = 1 theproofis
obvious.Now supposethatn > 1. Let A' be the matrixobtainedfromA by deletingthe nth
rowoperationswhichplacesA in reduced
column.We observethatanysequenceof elementary
VOL. 57, NO. 2, MARCH1984
93
ifB and C are
rowechelonformalso placesA' in reducedrowechelonform.Thusby induction,
in thenthcolumnonly.AssumeB 0 C. Then
reducedrowechelonformsof A, theycan differ
thereis an integer
j suchthatthejth rowof B is not equal to thejth rowof C. Let u be any
columnvectorsuchthatBu = 0. ThenCu= 0 and hence(B - C) u = 0. We observethatthefirst
n- 1 columnsof B - C are zero columns.Thus thejth coordinateof (B - C)u is (bj - c1t1)utl.
Sincebjl 0 CP,we musthaveu,,= 0. Thus anysolutionto Bx = 0 or Cx = 0 musthavex,,=0 . It
followsthatboth the nth columnsof B and C mustcontainleadingl's, forotherwisethose
choosethevalueofx,. Butsincethefirst
columnswouldbe freecolumnsand we couldarbitrarily
n - 1 columnsof B and C areidentical,therowin whichthisleading1 mustappearmustbe the
same forbothB and C, namelytherowwhichis thefirstzero rowof thereducedrowechelon
in thenthcolumnsofB and C mustall be zero,we have
formofA'. Becausetheremaining
entries
thetheorem.
B = C, whichis a contradiction.
Thisestablishes
Let A be an mnx
n
We remarkthatthisproofeasilygeneralizes
to thefollowing
proposition:
formsuch
matrixwithrowspace W. Thenthereis a uniquemx n matrixB in reducedrowechelon
thattherowspaceofB is W. (For anotherproof,see [2,p. 56].)
References
[1 ]
[2]
NewYork,1977.
LinearAlgebra,
Wiley,
H. Anton,Elementary
NewJersey,
1961.
Prentice-Hall,
Englewood
Cliffs,
K. Hoffman
andR. Kunze,LinearAlgebra,
and livesin thesea?
Riddle:Whatis non-orientable
-ROBERT MESSER
(See News and Lettersif you giveup.-ed.)
94
MATHEMATICS MAGAZINE