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LINEAR ALGEBRA

BERNARD KOLMAN DAVID R. HilL

 Preface xiii

The answers to all odd-numbered exercises appear in the back o f the book. An In-
structor's Solutions Manual (ISBN: 0-13-229655-1), containing answers to all
even-numbered exercises and sol utions to all theoretical exercises, is available (to
instructors only) from the publi sher.

PRESENTATION We have learned from experience that at the sophomore level, abstract ideas must
be introduced quite gradually and must be based on firm foundations. Thus we
begin the study of linear algebra with the treatment of matrices as mere arrays of
numbers that arise naturally in the solution of sys l ~ms o f linear equations, a prob-
lem al ready familiar to the studen1. Much al1cntion has been devoted from one
edition to the next to refining and improving the pedagogical aspects of the exposi-
tion. The abstract ideas are carefully balanced by the considerable emphasis on the
geometrical and computational aspects o f the subject. Appendix C, Illt roductioll
to Proofs can be used 10 give the studcnt a quick introduction 10 the foundations o f
proofs in mathematics. An expanded version o f this material appears in Chapter 0
of the Student Solutions Manual.

MATERIAL COVERED [n using this book, for a one-quaner linear algebra course meeti ng four times a
week, no difficulty has been e ncountered in covering eigenvalues and eigenvectors,
omil1ing thc optional matcrial. Varying the amount oftimc spent on the thcoretical
material can readily change the level and pace of the course. Thus, the book can
be used to teach a num ber of different types o f courscs.
Chapter I deals v.-ith matrices and their propcnies. In this chapter we also
provide an carly introduction to matrix transformations and an application of thc
dot product to statistics. Methods for solving systems o f lincar equations are dis-
cussed in Chapter 2. Chapter 3 introduces the basic properties of determinants
and some of their applications. In Chapter 4, we corne to a more abstract notion,
rcal vector spaces. Here we tap some o f the many geometric ideas that arise nat-
urally. Thus we prove that an II -dimensional, real vector space is isomorphic to
R", thc vector space of all ordered n-tu ples of real num bers. or the vector space
of all II x I matrices with real entries . Since R" is but a slight generalization of
R2 and R3. two- and three-dimensional space are discussed at the beginning of
the chapter. This shows that the notion of a finit e-dimensional. real vector space
is not as remote as it may have seemed when first introduced. C hapter 5 cov-
ers inner product spaces and has a strong geometric orientation. Chapter 6 deals
with matrices and linear transformations: here we consider the dimension theo-
rems and also appl ications to the solution of systems of linear equations. C hapter
7 considers eigenvalues and eigenvectors. In this chapter we completely solve the
diagona[ization problem for symmetric matrices. Chapter 8 (optional) p re~e n ts
an introduction to some applications of e igenvalues and eigenvectors. Section 8.3,
DOll/inalll Eigellvalue and Principal Compollent Analysis, hi ghlights some very
useful results in linear algebra. 11 is possible to go from Section 7.2 directly to
Section 8.4. Differelllial Equations. showi ng how linear algebra is used 10 solve
differcntial equations. Section 8.5. Dynamical Sy.flem.\· gives an application of lin-
ear algebra to an imponant area o f modern applied mathematics. In this chapter we
also discuss real quadratic fornl s, conic sections. and quadric surL1ces. Chapter
9. M ATLAB for Linear Algebra, provides an introduction to M ATLAB . Chapter
10. MATLAB Exercises. consists of [47 exercises that are designed to be solved
xiv Preface

using MATLAB. Appendix A reviews some very basic material dealing with sets
and functions. It can bc consulted at any time as needed. Appendix B, on com-
plex numbers, introduces in a brief but thorough manner complex numbers and
their use in linear algebra. Appendix C provides a brief introductio n to proofs in
mathematics.

MAnAS SOFTWARE The instructional M-filcs that have been developed to be used for solving thc ex-
ercises in thi s book, in particular those in Chapter 9, are available o n the follow-
ing website: ww w.prenhall.comlkolman. These M-files arc designed to transform
many of MATLAB'S capabilities into courseware. Although the computational
exercises can be solved using a number of software packages, in our judgment
MATLAB is the most suitable package for this purpose. MATLAB is a versatile
and powerful soft ware package whose cornerstone is its linear algebra capabili-
ties. This is done by providi ng pedagogy that allows the student to interact with
MATLAB. thereby letting the student think through all the steps in the solution
of a problem and relegating M ATL AB to act as a powerful calcu lator to relieve the
drudgery of tedious computation. Indeed, this is the ideal role for MATLAB (or any
other simi lar package) in a beginning linear algebra course, for in this course, more
than many others, the tedium of lengthy computatio ns makes it almost impossible
to solve a modest-size problem. Thus, by introduci ng pedagogy and reining in
the power of MATL AB, these M-files provide a working partnership between the
student and the computer. Moreover. the intrcxluction to a powerfu l tool such as
M ATLAB early in the student 's college career opens the way for other software
support in hi gher-level courses, especially in science and engineeri ng.
MATLAB incorporates professionally developed quality computer routines for
linear algebra computation. The code employed by M ATL AB is wrillen in the C
language and is upgraded as new versions of MATL AB arc released. MATLAB
is available from The Math Works Inc., 3 Apple Hi ll Dri ve, Natick, MA 01760,
e-mail: info@mathworks.com. [508-647-70001. The Student version is available
from The Math Works at a reasonable cos\. This Sllldent Edition of MATLAB
also includes a version of Maple™, thereby providing a symbolic computational
capability.

STUDENT SOLUTIONS The Student Solutions Manual (ISB N: O-13-229656-X), prepared by Denni s R.
MANUAL Kletzi ng, Stetson Uni versity, contains sol utions to all odd-numbered exercises,
both nu merical and theoretical.

ACKNOWLEDGMENTS We arc pleased to express our thanks to the following reviewers of the first eight
editions: the late Edward Norman. University of Central Florida; the late Charles
S. Duris, and Herbert J. Nichol, both at Drexel University; Stephen D. Kerr, We-
ber State College; Norman Lee, Ball State University; William Briggs, University
of Colorado: Richard Roth. Uni versity of Colorado; David Stanford , College of
William and Mary; David L. Abrahamson, Rhode Island College; Ruth Berger,
Memphis State University; Michael A. Geraghty, University of Iowa; You-Feng
Lin. University of South Florida; Lothar Redlin, Pennsy lvania State University,
Abington; Richard Sot, University of Nevada, Reno; Raymond Southworth, Pro-
fesso r Emerillls, College of William and Mary ; 1. Barry TUTen , Oakland Univer-
sity : Gordon Brown, University of Colorado; Mall Insall, Uni versity of Mis;ouri
 Preface xv

at Rolla; Wolfgang Kappe State University of New Yo rk at Bing hampton ; Richa rd

P. Kubelka, San Jose State University; James Nation, Uni ve rsit y o f Hawaii ; David
Peterson, Uni versity o f Central Arkansas; Malcolm J. Sherman, State Uni ve rsity
o f New York at Albany: Alcx Stanoyevi tc h, Uni versity of Hawaii; Barbara Tabak,
Brandeis Uni ve rsity; Lorin g W. Tu Tufts University; Manfred Ko lster. Mc Master
University ; Daniel Cunningham, Buffalo State College; Larry K. Chu. Mino t State
Unive rsity : Danie l King, Sarah L1wre nce University; Kevin Yang, Minot State
Unive rsity ; Avy Soffer, Rutgers University; David Kaminski, Unive rsity of Leth-
bridge, Pat ricia Beaulieu. Uni versity o f Louis ia na, Will Murray, California State
Um versi ty at Long Beach and o f the ninth edition: Thalia U . Je llres. Wic hita State
Uni versi ty, Ma nuel Lopez. Rochester Institute of Technology, Thomas L. Scofield.
Calvin Coitege, Jim Gehnnann, California State Uni versity, Sacramento, Jo hn M.
Erdman. Po rtland State University. Ada Che ng, Kette rin g Uni versity. Jue rgen Ger-
lach, Radfo rd Uni vers ity, and Martha Allen, Georgia College and State Uni versity.
The nume rous suggestions, comme nts, and criticisms of these people greatly
improved the manuscript.
We thank De nnis R. Kletzing, who typeset the e ntire manuscrip t, the Student
Solutions Manl/al. a nd the Instructor's So/utions Mal/ual. He found and corrected
a nllmhe r of mathemari cal e rrors in the manusc ripr . Ir was a pleas llfe workin g with
him.
We thank Blaise deSesa for carefully checki ng the answers to all the exercises.
We thank William T. Williams. fo r kind ly letting us put his striking image
Tran e o n the cover o f this editio n.
We also thank Lilian N. Brady and Nina Edelman, Te mple Uni versity, for crit-
ically and carefully reading page proofs: a nd instructors and students from many
institutions in the United Slates and other countries, fo r sharing with us their ex-
pe rie nces with the book fo r the past 38 years and offering helpful s uggestio ns.
Finally, a sincere expressio n o f thanks goes to Scot! Di sanno. Senior Ma n-
aging Editor: to Holly Stark, Senior Edito r; to Jennifer Lonschein. Editorial As-
s istant, and to the e nti re staff of Prentice Hall for the ir e nthus iasm , interest. and
unfailing cooperation during the conceptio n, design, productio n. and marketing
phases o f this editio n. It was a genuine pleasure wo rki ng with the m.

B.K.
D.R.H.
This page illlenriollally left blank
 THE
This course may be unlike any other mathematics course thaI you have stud ied
thus far in at least two important ways. First, it may be you r initial introduction
to abstraction. Second, it is a mathematics course that may well have thc greatest
impact on your vocation.
Unlike other mathematics courses, this course will not give you a toolkit o f
isolated computational techniques for solvi ng certain types of problems. Instead.
we will develop a core of material called linear algebra by introducing certain def-
initions and creating procedures fo r determining properties and proving theorems.
Proving a theorem is a "kilt that lakes time to master, so we will develop your skill
al proving mathematical results very carefully. We introduce you to abstraction
slowly and amply illustrate each abstract idea with concrete numerical examples
and applications. Although you will be doing a lot of computations, the goal in
most problems is not merely to get the "right" answer. but to understand and be
able explain how to get the answer and then interpret the result.
Linear algebra is used in the everyday world to solve problems in other areas
o f mathematics. physics. biology, chemi stry. engineering, statistics. economics, fi-
nance. psychology, and sociology. Applications that use linear algebra include the
transmission of information, the development o f special effect s in film and video.
recording of sound, Web search e ngines on the Internet, global positioning system
(GPS) and economic analyses. Thus, you can see how profoundly linear algebra
affects you. A selected number of applications are included in this book, and if
there is enough time, some of these may be covered in your course. Additionally,
many of the applications call be used as sel f-study projects. An extensive list of
;Jrplic:lIions npPC;Jrs in thc front insiflc COVC L
T here arc four different types of exercises in this book. First, there are com-
putational exercises. These exercises and the numbers in them have been carefull y
chosen so that almost all of them can readily be done by hand . When you use
linear algebra in real applications, you will lind that the problems are much bigger
in size and the Ilumber,; that occur in them arc not always "nice." Thi s is not a
problem because you will almost certainly use powerful software to solve them. A
taste of this type of software is provided by the third type of exercises. These are
exercises designed to be solved by using a compmer and M AT LAB TM, a powerful
matrix-based application that is widely used in industry. The second type of ex-
ercises are theoretical. Some o f these may ask you to prove a result or discu;s an
idea. The fourth ty pe of exercises are discussion exercises, which can be used as
group projects. In today's world, it is not enough to be able to compute an answer;
you o fte n have to prepare a report di scussing your solution, justifyi ng the steps in
your solution. and interpreting your results. These types o f exercises will give you
experience in writing mathematics. Mathematics uses words. not just symbols.

xvii
xviii To the Student

How to Succeed in Linear Algebra

• Read the book slowly with pencil and paper at hand. You might have to read
a particular section more than once. Take the time to verify the steps marked
"verify" in the text.
Make sure to do your homework on a timely basis. If you wait unti l the prob-
lems are explained in class, you will miss learning how to solve a problem by
yoursclf. Even if you can't complete a problem. try it anyway, so that when
you see it done in class you will understand it more easily. You might find
it helpful to work with other students on the material covered in <:lass and on
some homework problems.
Make sure that you ask fo r help as soon as something is not clear to you. Each
abstract idea in this course is based on previously developed ideas- much like
laying a foun dation and then bui lding a house. If any o f the ideas are fuzzy to
you or missing, your knowledge of the course will not be sturdy enough for
you to grasp succeeding ideas.
Make usc of the pedagogical tools provided in this book. At the e nd of each
section in the fi rst eight chapters. we have a list of key terms; at the end o f each
o f thc first seven chapters we have a chapter review, supplementary exercises,a
chapter quiz, and discussion exercises. Answers to the odd-numbered compu-
tational exercises appear at the end of the book. The Student Solutions Manual
provides detailed solut ions to all odd-numbered exercises, both numerical and
theoretical. It can be purchased from the publisher (ISBN O-13-229656-X).
We assure you that your efforts to learn linear algebra well will be amply
rewarded in other courses and in your professional career.

We wish you much success in your study of linear algebra.

CHAPTER

Linear Equations
and Matrices

m Systems of Linear Equations

One of the most frequently recurring practical problems in many field s of study-
such as mathematics. physics, biology, chemistry, economics. all phases of engi-
neering. operations research, and the social sciences- is that of solvi ng a system
of linear equations. The equation
(I )

which expresses the real or complex quantity b in tcrms of the unknowns X I , X2,
.. • , X" and the real or complex constants (II. (12 • ... • (In, is called a linear equa-
tion. In many applications we are given b and must find numbers Xl. Xl . ... . X"
satisfying (I).
A solution to Iinea.r Equation (I) is a sequence of /I numbers SI.,f2.. .,f".
which has the propeny that ( I) is satisfied when XI = ,fl, -'"2 = S2, . .. ,X" = s" are
substituted in (1). Thus XI = 2, X2 = 3. and XJ = - 4 is a solution to the linear
equation

because
6(2) - 3(3) + 4( - 4) ~ - 13.
More generall y, a system of III linear equations in /I unknowns, Xl, -'"2 .. . .. X".
or a linear system, is a set of III linear equations each in n unknowns. A linear

NOIe: Appendix A revitws some \"Cry basic material dealing witll sets and functions. It rail be
consulted at any time. as needed.
I
2 Chapler 1 Linear Equations and Matrices

system can conveniently be wri tten as

(2)

Th us the ith equation is

I.n (2) the (Ii) are known constants. Given values of hi. h 2 • ...• h",. we want to find
values of XI. X2 . .. .. x " that will satisfy each equation in (2).
A solution to linear system (2) is a sequence of 1/ numbers SI. S2. . . SII'
which has the property that each equation in (2) is satisfied when XI = .1"1. X2 = .\·2 ,
.... x" = .f" are substituted .
If the linear system (2) has no solution, it is ~aid to be inconsistent: if it has
a solution. it is called consistent. If b l = h2 = = b", = 0, then (2) is
called a homogeneous system. Note that XI = x? = = x" = 0 is always
a solution to a homogeneous system; it is called the trivial solution. A solution
to a homogeneous system in which not all of XI. X2 • ... • XII are zero is called a
nontrivial solution.
Consider another system of r linear equations in 1/ unknowns:

CIIXI + CI2X2 + ... + Cl n T" = til

C21XI + CnX2 + ... + Cil,T" = til

(3)

We say that (2) and (3) are equivalent if they both have exactly the same solutions.
The linear system
EXAMPLE 1
XI - 3X2 = - 7
(4)
2xI + X2 = 7

has only the solution XI = 2 and X2 = 3. The linear system

8x, - 3X2 = 7
3x, - = 0
2tl (5)
IOXI - 21:2 = 14

also has only the solution XI = 2 and X2 = 3. Thus (4) and (5) are equivalent. •
To find a solutio n to a linear system, we shall use a technique called the
method of elimination: that is, we eliminate some variables by adding a multiple
of one equation to another equation. Eliminatio n merely amounts to the develop-
ment of a new linear system that is equivalent 10 the original system, but is much
simpler 10 solve. Readers have probably confined their earlier work in this area to
 1.1 Systems of linear Equations 3

linear systems in which III = II. that is, li near systems having as many equations
as unk nowns. In this course we shall broaden our outlook by dealing with systems
in which we have 11/ = n, 11/ < II. and 11/ > n. Indeed. there are numerous applica-
tions in which III i= II. If we deal with two, three, or four unknowns, we shall often
write them as x, y, Z, and w. In this section we use thc method of eliminatio n as it
was studied in high school. In Section 2.2 we shall look at this method in a much
more systematic manner.

The director of a trust fund has $100.000 to invest. The rules of the trust state
EXAMPLE 2
that both a certificate of deposit (CD) and a lo ng- te rm bond must be used. The
director's goal is to have thc trust yield $7800 on its investments for the year.
The CD chosen returns S% per annum. and the bond 9% . The director determines
the amount x to invest in the CD and the amount y to invest in the bond as follow s:
Since the total inve:;tment is $100,000, we must have x +)' = 100.000. Since
the desired return is $7800. we obtain the equation O.OSx + 0.09)' = 7800. Thus,
we have the linear system

x + )' = 100.000
(6)
O.OSx + 0 .09)" = 7800.

To eliminate x, we add (-O.OS) times the first equation to the second, obtaining

0.04)' = 2800.

an equation having no .r term. We have eliminated the unknown x. Then solving

for y . we have
y = 70.000.

and substituting into the lirst equation of (6), we obtain

x = 30.000.

To check that x = 30.000, Y = 70.000 is a solution to (6), we verify that these

values o f x and )' satisfy each of the equations in the given linear system . Thi s
solutio n is the only solutio n to (6): the system is consistent. The director of the
trust should invest $30,000 in the CD and $70,000 in the long-term bond . •

Consider the linear system

EXAMPLE 3
x -3)"=-7
(7)
2x - 6y = 7.

Agai n. we decide to eliminate x. We add (-2) times the fi rst equation to the
second one, obtai ning
0 = 21.
which makes no sense. This means that (7) has no solutio n: it is inconsistent We
could have come to the same conclusion from observing that in (7) the left side of
the second equation is twice the len side of the fi rst equatio n, but the right side of
the second equation is not twice the ri ght side of the fi rst equation . •
4 Chapler 1 Linear Equations and Matrices

Consider the linear system

EXAMPLE 4
x + 2)' + 3z = 6
2x - 3)' + 2z = 14 (8)
3x + )' - z = - 2.

To eliminate x, we add (- 2) times the first equation to the second one and (- 3)
times the fi rst equation to the third one, obtaini ng

- 7)' - 4 z= 2
(9)
- 5)' - IOz = - 20.

Th is is a system of two equations in the unknow ns.v and z. We multiply the second
equation of (9) by (- tl. yieldi ng

- 7y - 4z= 2
)' + 2z = 4,
which we write. by interchanging equations. as

y + 2z= 4
( 10)
- 7y - 4z= 2.

We now eliminate y in ( 10) by adding 7 times the first equation to the second one,
to obtai n

10z = 30.

z= 3. ( II )

Substituting this value of z into the first equation of (10), we find that y = - 2.
Then substituting these values of y and z into the first equation of (8), we find that
x = I. We observe further that our elimi nation procedure has actually pnxluced
the linear system

x + 2y + 3:; = 6
)' 1 2;: = 4 (12)
:; = 3,
obtained by using the first equations of (8) and (10) as well as (II). The importance
o f this procedure is thaI, although the linear systems (8) and (12) are equi valent.
(12) has the advantage that it is easier to solve. •

Consider the linear system

EXAMPLE 5
x + 2)' - 3z = - 4
(13)
2x + y - 3z = 4.
 1 .1 Systems of linear Equations 5

Eliminating x, we add (-2) times the first equation to the second equation to get

-3)' + 3z = [ 2. ( 14)

We must now solve (14). A solution is

)' =z - 4.
where z can be any real number. The n from the fi rst equation of ( [ 3),
x = -4 - 2y + 3;:
= - 4 - 2(z - 4) + 3z
= z + 4.
Thus a solution to the linear system (13) is

x = z +4
Y= z- 4
z = any real number.
This means that the linear system (13) has infinitely many solutions. Evcry time
we assign a value to z we obtain another solution to ( 13). Thus, if z = I, then

x = 5. y = - 3. and

is a solution, while if z = - 2. then

.r = 2, y = - 6. and z = -2

is another solutio n.
•
These examples suggest that a li near system may have a unique solution, no
solution, or infi ni tely many solutions.
Consider next a li near system of two eq uations in the unknowns x and y:

(JIX + (J2)' = el
(15)
blx + b 2)' = C2 ·

The graph o f each o f these equations is a straight line. which we denote by il and
£2, respectively. If x = .\'1,)' = .f! is a solution to the linear system (15). then the
point (.1'1. S2) lies on both lines i l and i 2 . Conversely, if the point (.fl. Sl) lies on
both li nes i l and £2, then x = .1"[, )' = .\. ! is a solution to the linear system (15).
Thus we are led geometrically to the same three possibilities mentioned previously.
See Figure I. [.
Next. consider a linear system o f three equations in the unknowns x. y, and z:

([IX + b l )' + elZ = d l

(J2X + b 2 J' + elZ = d 2 ( 16)
(l3X + h3)' + e3Z = d 3 .

The graph of each o f these equations is a plane. denoted by PI. P2, and P3, re-
specti vely. As in the case of a linear system of two equations in two unknowns.
6 Chapler 1 Linear Equation s and Matrices

(a) A unique solution. (b) No solution. (e ) Infinitely many solutions.

)' )' )'

I,
I,
- 7"+---'__- < ---\1--\--" ---+7"'---- "
I, I,
FIGURE 1. 1

the linear system in (16) can have infini tely many solutions, a unique solution, or
no solution. These situations are illustrated in Figure 1.2. For a more concrete
ill ustration of some of the possible cases, consider that two intersecting walls and
the ceiling (planes) of a room intersect in a unique point, a corner of the room,
so the li near system has a unique solution. Next, think of the planes as pages of
a book. Three pages of a book (held open) intersect in a straight line, the spine.
Thus, the linear system has infi ni tely many solutions. On the other hand, when the
book is closed, three pages of a book appear to be parallel and do not intersect, so
the linear system has no solution.

(a) A unique solution. (b) Infinitely many solutions. (c) No solution.

p,/ P,

V
./
p;/
FIGURE 1.2

If we exami ne the melhod of elimination more closely, we fin d that it involves

three manipulations that can be perfonned on a linear system to convert it into
an equivalent system. These manipu lations are as follows:
I. Interchange the ith and Jth equations.
2. Multiply an eq uation by a nonzero constant.
3. Replace the ith equation by c times the jth equation plus the ith equatio n,
i i= j. That is, replace

by
 1.1 Systems of linear Equations 7

It is not difficult to prove that performing thes~ manipulations on a linear sys-

tem leads to an equivalent system. The next example proves this for the second
type of manipulation. Exercises 24 and 25 prove it fo r the firsl and third manipu-
lations, respectively.

Suppose that the ith equation of the linear system (2) is multiplied by the nonzero
EXAMPLE 6
constant c. producing the linear system

(lIIXI + {/12X2 +
anxi + {/ 22 X l +
(17)

If XI = .f1 • ..I:2 = .f2 . . . . . x" = .1'" is a solution to (2), then it is a solution to all the
equations in (17), except possibly to the ith equation . For the ith equation we have
c(a jl S1 + (/i2.f2 + ... + (linS,,) = Cbi

CUi 1St +Clli2.f2 + ... + Cll,,,S,, = Chi.

Thus the ith equation of (17) is also satisfied. Hence every solution to (2) is also
a solution to (17). Conversely, every solution to (17) also satisfies (2). Hence (2)
and (17) are equivalent systems. •

The following example gives an application leading to a linear system of two

equations in three unknowns:

(Production Planning) A manufacturer makes three different types of chemical

EXAMPLE 7
products: A, B. and C. Each product must go through two processing machines:
X and Y. The products require the following times in machines X and Y:
I. One ton of A requires 2 hours in machine X and 2 hours in machine Y.
2. One ton of B requires 3 hours in machine X and 2 hours in machine Y.
3. One ton of C requires 4 hours in machine X and 3 hours in machine Y.
Machine X is available 80 hours per week. and machine Y is available 60 hours
per week. Since management docs not want to keep the expensive machines X and
Y idle, it would like 10 know how many tons of each product to make so that Ihe
machines are fully utilized. It is assumed that the manuL1cturer can sell as much
o f the products as is made.
To solve this problem. we let XI> Xl, and X) denote the number of toilS of
products A , B , and C. respectively. to be made. The number of hours that machine
X will be used is
2xI + 3X2 + 4X 3 .
which must equal 80. Thus we have

2xI + 3X2 + 4X 3 = 80.

8 Chapler 1 Linear Equation s and Matrices

Similarly, the number of hours that machine Y will be used is 60, so we have

2xI + 2X2 + 3X3 = 60.

Mathematically, our problem is to fi nd nonnegati ve values of XI. X2, and X3 so that

2xI + 3X2 + 4x] = 30

2{1 + 2x2 + 3x] = 60.

This li near system has infinitely many solutions. Followi ng the method of
Example 4, we sec that all solutions are given by
20 - x]
XI = --2--

X2 = 20 - X3

X] = any real nu mber such that 0 S x ] S 20,

since we must have X I ::: 0 ,","2::: 0, and X3 .:::. O. When x ] = 10, we have
= 5. X2 = 10, x ] = 10
"
while
x, ~
" X2 ~ 13, x] ~ 7

"
when X 3 = 7. The reader should observe that o ne solution is just as good as the
other. There is no best solution un less addi tional infonnation or restrictions are
given. •
As you have probably al ready observed, the method of eli mination has been
described, so far, in general terms. Thus we have not indicated any rules for select-
ing the unknowns to be eliminated . Before providing a very systematic descrip-
tion of the method of elimi nation, we introduce in the next section the notion of
a matrix. This will greatly simpli fy our notational problems and will enable us to
develop tools 10 solve many imponant applied problems.

Key Terms
Linear equation Consistent system Unique solution
Solulion of a linear equation Homogeneous system No solution
Li near system Trivial solution Infinitely many solutions
Unknowns Nontri vial sol ution Manipulations on linear systems
Inconsistent system Equivalent systems Method of elimination

M.M Exercises

I II EJerdse.\· Ilhro l/gh 14. .\"O/re each gil'ell lillear ~)'~'Ielll by 3. 3x + 2y + : = 2 4 . .{ +y =5

fhe It1nl/Od of dimillmivil. 4x + 2y + 2: = 8 3x+3y=IO
x - y+ : =4
l. x + 2)' = 8 2. 2x-3y+4;= - 12 5. 2{+4y+6Z = - 12 6. x+ y-2: =S
h - 4y =4 x - 2y + : = -5 2{ - 3y-4~ = 15 2x + 3y + 4: = 2
3x+ y+2z = 3x+4y +5: = -8
 1 .1 Systems of linear Equations 9

7. .1 + 4y - :: = 12 8. 3x + 4)' - z =8 (a ) Verify that Xl = I. Yl = - 1. Zl = - I is a solution.

h + 8y - 2;:: = 4 6_1 + 8)' - 2;:: = 3 (b) Verify that X2 = - 2.)'2 = 2. Z2 = 2 is a sol ution.
9. x + ), +3z =12 10. x+ )' =1
2x + 2)' + 6;:: = 6 2, - ), =5 (e) Is x = X l + X2 = - I. Y = Yl + Y2 = I. and
Z = Zl + Z2 = 1 a solution to the linear system?
3x + 4)' = 2
II . h+3y=13 12. x - 5)' = 6 (d ) Is 3x. 3y. 3z . where x. y. and; are as in part (c). a
x - 2)' = 3 1x + 2)' = I solution to the linear system?
5:.: + 2)' = 27 5.{ +
2)' = I 20. Without using the method of elimination . so lve the linear
13. x + 3.1' = - 4 14. 21 + 3)' - z = 6 system
2x+5y=-8 2, - ),+2z = - 8 2x + )" - 2z = - S
x+3y=-S 3x - y+ z = - 7 3y + z = 7
15. Given the linear system 4.
2x - ) =5 21. Without using the method of el imination. solve the linear
4x -2) = /. system
4x 8
(a ) Delennine a particular vahle of I so thallhe system
is consistent. -2\" + 3)' = - 1
(b) Detennine a particular value of f so thallhe system 3x+Sy - 2z = II.
is mconslstenl. 22. Is there a value ofr so that X = I. Y = 2. Z = r is a
(c) How many different values of I can be selected in sol ution to the followin g linear system? If there is. find
pari (b)? it.
16. Given the linear system 2x + 3)' - Z = I I
x - y+2:=-7
3x+4)=s 4x+ y-2z = 12.
6x+8) =I.
23. Is there a val ue ofr so that x = r.y = 2. : = lis a
(a) Determine particular values for l' and I so Ihal the
sol ution to the follol'.-ing linear system? If there is. find
system is consistent.
it.
(II) Dtlennine p.uticulal values fOJ 1 and I so thai the 3x - 2z= 4
system is inconsistent.
x- 4)' + z =-S
(c) What relationship between the values of s and I will
-2x + 3y + 2z = 9.
guarantee Ihat Ihe system is consistent?
17. Given the linear system 24. Show that the linear system obtained by interchanging
two equations in (2) i, equ ivalent to (2).
x+ 2y=10
3x + (6+1»), = 30. 25. Show that the linear system obtained by adding a multi-
ple of an equation in (2) to another equation is equil'alent
(a ) Determine a particular value of I so that the system to (2).
has infinitely man y solutions.
26. Describe the number o f points that simultaneously lie in
(b ) Determine a particular value of I so that the system
each of the three planes shown in each part of Figure 1.2.
has a unique solution.
(e) How m:my different values of I can be selected in 27. Describe the number of points that simultaneously lie in
part (b)? each of the three planes shown in each part of Figure 1.3.
18. Is every homogeneous linear system always consistent? 28. Let C l and C2 be circles in the plane. Describe the num-
Explain. ber of possible points of intersection of C 1 and C2 • illus-
19. Given the linear system trate each case with a figure.
29. Let Sl and 52 be spheres in space. Describe the number
2.1"+3), - z =O
of possible points of intersection of Sl and S2. Il lu5lrate
x - 4)' +5z =0. each case with a figure.
10 Chapter 1 li near Equations and Matrices

/ p, / avai lable 16 !lours per day, how many tons of eac h type

/ p,
/ p,
(,)
/
/ OJ1':

(b)
P,
of de\'elopcr can be p-oduced so that the plants are fu ll y
U."ed?
34, Suppose that the three points ( I . - 5). (- I. I) . and (2. 7)
lie on the parabola pel) = (I.r 2 + bx + c.
(a) Determine a linear system of thre e equations in three
unknowns that must be solved to find {/. h. and Co

(b ) Solve the linear system obtained in part (a) for (I, b,

and c.
35, An inheritance of S24,000 is to be divided among three
truStS, with the second trust receiving twice as moch as
the first trust. The three trusts pay interest annually at
the rales o f 9%, 10%. and 6%, respectively. and return a
total in interest of 52210 at the end of the first ye ar. How
mu ch was invested in each trust?
(e) • 36. For the wftware you are using. determine the command
that "automatically" wives a linear sy~tem of eqllations.
FIGURE 1.3
• 37, Use the command from Exercise 36 to solve Exercises 3
3 0. :\n oil refi nery produces low sulfur and high s ulfur fueL and 4. and compare the Output with the resulL~ yoo ob-
Eac h IOn of low-sulfur fuel requires 5 minutes in the tained by the method o f elimination.
blendi ng plam and 4 minules in the refining plant: each
ton of high.sulfur fuel require; 4 minutes in the blending .!. 38. Solve the linear system
plam and 2 minutes in the refining planl. If the blend- x+h'+ ~z =
109 plant is available for 3 hours and lhe refining plant is
available for 2 hours. how many tons o f each lype of fuel tx + jy+ ;z= N
should be manufactured so that the plants are fully used ? f x+ !y+ ~ z= ~
3 1. A plastics manufacturer m:lkes two types of plastic: reg- by using YOllr software. Cornp.tre the com puted soilltion
ular and special. Each ton of regular pl3SIic requires 2
houo: in plam A :md 5 hours in planl B: eac h ton of ~pe_
with the exact solution x = !,
y = ~. z I. =
c;:l1 plastic requires 2 hours In plam A and 3 hours in .!. . 39. If your software includes acce§s to a computer algebra
sy.~l em (CAS), use it as follows:
piam B. Jfp lant A is avai lable [I hours per day and plant
B is available 15 hours per day, how many tons of each (a) For the linear sys tem in Exercise 38. replace the
type of plastic can be made daily so thm the plants are fraction ~ with its decimal equivalent 0.5. Enter this
fully used? sys tem into your software and use the appropriate
32. A dietician is preparing a meal consisti ng of foods A. B. CAS commands to solve the system. Compare the
and C. Each ounce of food A contains 2 units of pro_ solution with that obtained in Exercise 38.
lein. 3 units of fal. and 4 units of carbohydrate. Each (b) In some CAS environments you ca n select the num-
ou nce of food B contains 3 units of prote in. 2 units of ber of digits to be used in the calculations. Perfonn
fn1. and 1 unit of carbohydrme. Each ounce of food C p:1I1 (a) with d i.!!it choices 2, 4. and 6 to see what
contains 3 unils of protein. 3 units of fm. and 2 units of infillence such selections have on the computed so·
carbohydrate. If the meal IllUi t provide exactly 25 units lution.
of protein. 24 units of fat. and 2 I units of carbohydrate. .! 40, If )'our software includes acce.~s to a CAS and Y01 can
how many ounces of eac h lype of food should be used ? ...elect the number of digits used in calc ulations. do the
33. A manufacturer makes 2-minule. 6-minute. and 9·minute following: Enter the linear system
fi lm deve lopers. E.1Ch ton of 2·minute developer requires
6 minutes in plant A and 24 minutes in plant B. Each ton
0.7I.1 + 0.21), = 0 .92
of 6·minute developer require s 12 mi nutes in plant A and 0.23x + 0.58)' = 0 .8 1
12 minutes in plant B. Each too of9·minute de\'eloper reo into the program. Have the software solve the syste m
quires 12 minutes in plant A and 12 minutes in plant B. with digit choices 2. 5, 7. and 12. Briefly discus. an y
If plant A is available 10 hours per day and plant B is variations in the wlutions generated.
 1.2 Matrices 11

m Matrices
If we exami ne the method of elimination described in Section 1.1, we can make Ihe
following observation: Only the numbers in front oftn e unknowns XI, X2, . . . , x"
and the numbers hi. b 2 • ...• b m on the right side are being changed as we perform
the steps in the method of elimination. Thus we might think o f looking fo r a way
of writing a linear system without having to carry along thc unknowns. Matrices
enable us to do this- that is, to write linear systems in a compact form that makes
it easier to automate Ihc elimination metho d by using computer software in orde r
to obtain a fast and efficient procedure for findin g solutions. The usc o f matrices.
however, is not merely that of a convenie nt notation. We now develop operations
on matrices and will work with matrices according to the rules they obey: this will
e nable us to solve systems of linear equations and to handle other computational
problems in a fast and effici ent manner. Of course, as any good defini tion should
do. the notion o f a matrix not only provides a new way of looking at old problems.
but also gives ri se to a great many new questions. some of which we study in this
book.

A n 11/ X 11 matrix A is a rectangular array of mil real or complex numbers arranged

DEFINITION 1.1
in //I hori zontal rows and II vertical columns:
a!! al2
a2! a22

(I)
.....- ith row

•L- jth column

The ith row o f A is

ai ll ] (I :;: i '=:: /I/ );

thejth column of A is

aa,'i] (l .=::J.=:: n).

[ ,L
We shall say that A is III by II (written as III x II ). If 11/ = II , we say that A is a
square matrix of order II , and thai the numbers II!!. a22 . .... all" form the main
diagonal of A. We rc fer to the number a ij , which is inlhe ith row and Jth column
o f A, as the i,j th element of A, or the (i.j) entry of A. and we often write (I) as
12 Chapter 1 Li near Equations and Matrices

Lei
EXAMPLE 1

A= [ [
- \
2
0 n B= [ I+ i
2 - 3i
4;
- 3 .
1 C ~ Hl
D~ [i 0
- \ n E ~ [3]. F ~ [- \ 0 2] .

T hen A is a2 x 3 matriK witha l2 = 2,{/ u = 3,(122 = 0, and un = [; B is a2 x 2

matrix withb ll = I +i,bl2 = 4i. b2 1 = 2 - 3i.and b n = - 3; e is a3 x I matrix
with C I I = L C2 1 = - I, and C3 1 = 2: D is a 3 x 3 matrix: E is a I x [ matrix:
and F is a [ x 3 matrix . I.n D, the elements dll = l. d 22 = 0, and d 3J = 2 form
the mai n diagonal. •

For convenience, we focu s much o f our attention in the illustrative examples

and exercises in Chapters [ -6 on matrices and e.\pressions containi ng only real
numbers. Complex numbers make a brief appearance in Chapter 7. An introduc-
tion to complex numbers, thei r propenies, and examples and exercises showing
how complex numbers arc used in linear algebra may be foun d in Appendix B.
An /I x I matrix is also called an lI·vector and i:; denoted by lowercase boldface
letters. When II is understood, we refer to II-vectors merely as vectors. Vectors
arc discussed at length in Section 4.1.

EXAMPLE 2 •
T he II-vector all of whose entries are zero is denoted by O.
Observe that if A is an II x /I matrix, then the rows of A are I x II matrices and
the columns of A are II x I matrices. The set o f all /I-vectors with real entries is
denoted by R". Simi larly, the set of all II-vectors with complex entries is denoted
by en.
As we have already pointed out. in the first six chapters o f this book we
work almost entirely with vectors in R" .

(Tabular Display of Data ) T he following matrix gives the airli ne distance;; be-
EXAMPLE 3
tween the indicated cities (i n statute miles):

London Madrid New York Tokyo

London 785 3469
5959 ]
Madrid [ 785
0 0 3593 6706
New York 3469 3593 0 6757
Tokyo 5959 6706 6757 0
•
(Production) Suppose that a manufacturer has fou r plants, each o f which makes
EXAMPLE 4
three products. If we let aU denote the number of units of product i made by planl
Another Random Scribd Document
with Unrelated Content
prisoners of war. This German general, executing the orders of the
German High Command—particularly of Keitel and Jodl—said that
those wounded men were to be treated as francs-tireurs and to be
delivered to the SD or to the Abwehr. Those wounded men were
turned over to the German Police and tortured and killed without
trial.
According to the statement of Goldberg, which I have submitted,
any man turned over to the SD was executed. Events took place on
21 June 1944 as indicated by Goldberg, “Twelve suspects were
arrested and turned over to the SD.”
Under the date of 16 August 1944, Page 133, this general of the
German Army had 40 men murdered after the battles at Bourg-
Lastic and at Cosnat:
“In the course of operation Jesser, on 15 July 1944 in the
Bourg-Lastic region, 23 persons were executed. Martial law.
Attack on Cosnat; 3 kilometers east of St. Hilaire, during
the night of 17 July, 40 terrorists were shot.”
On Page 136, this German general admits in his own diary that
our comrades were fighting as soldiers and not as assassins. This
general of the German Army acknowledges that the French Forces of
the Interior took prisoners:
“Southeast of d’Argenton, 30 kilometers southwest of
Châteauroux, the ‘Jako’ discovered a center of terrorists; 16
German soldiers were liberated; arms and ammunition were
captured; 7 terrorists were killed, 2 of them being captains.
One German soldier was seriously wounded.”
Another similar incident is also related further on:
“Discovery of two camps of terrorists in the region of
d’Argenton. Nine enemies were killed, two of whom were
officers; 16 German soldiers were liberated.”
At the bottom of the page he states, “We liberated two SS men.”
These French soldiers were entitled to the respect of their
adversaries. They conducted themselves as soldiers; they were
assassinated.
THE PRESIDENT: We will adjourn now until two o’clock.
[The Tribunal recessed until 1400 hours.]

Afternoon Session
MARSHAL: May it please the Court, I desire to announce that the
Defendants Kaltenbrunner and Seyss-Inquart will be absent from this
afternoon’s session on account of illness.
M. DUBOST: We had arrived, gentlemen, at the presentation of
the terrorist policy carried out by the German Army, Police, and SS,
indistinguishably united in their evil task against the French patriots.
Not only the militant patriots were to be the victims of this terrorist
policy. There were threats of reprisals against their relatives, and
these threats were carried into effect.
We submit Document 719-PS as Exhibit Number RF-406, which
you will find on Page 147 of the document book. It is the copy of a
teletype from the German Embassy in Paris to the Ministry of Foreign
Affairs in Berlin. The German Ambassador reports a conversation
which the Vichy unit had had with Laval.
The author of this message, who is probably Abetz, explains that
Bousquet, who was with Laval at the time of this conversation,
stated that he was completely ignorant of the recent flight of
Giraud’s brother:
“Madame Giraud, three of her daughters, her mother,
another brother and the daughter-in-law of Giraud, were in
Vals-les-Bains. I replied that such measures were
insufficient and that he must not be surprised if the German
police some day reverted to sterner measures, in view of
the obvious incompetence of the French police in numerous
cases.”
The threat was put into execution. We have already stated that the
family of General Giraud were deported.
We submit Document F-717 under Exhibit Number RF-407, Page
149 of your document book: “Paris, 1030 hours, 101, Official
Government Telegram, Paris, to the French Delegation of the IMT
Nuremberg.”
From this telegram it is evident that 17 persons, members of the
family of General Giraud, were deported to Germany. Madame
Granger, daughter of General Giraud, aged 32, was arrested without
cause in Tunis in April 1943, as well as her four children, aged 2 to
11 years, with their young nurse, and her brother-in-law, M.
Granger. The family of General Giraud was also arrested, on 9
October 1943. They were first deported to Berlin, then to Thuringia.
May I ask the forbearance of the Tribunal; the telegraphic style
does not lend itself to interpretation, “Sent first to Berlin and then to
Thuringia; women and children of M. Granger to Dachau.” (I
suppose that we must understand this to mean the wife of M.
Granger and the nurse who accompanied her.)
THE PRESIDENT: M. Dubost, what is the document?
M. DUBOST: This is a French official telegram. You have the
original before you, Mr. President, “—101—Official State Telegram
Paris,” typed on the text of the telegram itself.
THE PRESIDENT: Can we receive a telegram from anybody
addressed to the Tribunal?
M. DUBOST: Mr. President, it is not addressed to the Tribunal; it
is addressed to the French Delegation. It is an official telegram from
the French Government in Paris, “Official State Paris,” and it was
transmitted as an official telegram.
THE PRESIDENT: What does “IMT Paris” mean?
M. DUBOST: The International Military Tribunal in Paris. It is our
office in Paris at Place Vendôme—it is an office of the French
Ministry of Justice. The telegram begins, “General Giraud.” It is a
telegraphic declaration. The letters “OFF” at the beginning of the
telegram mean “Official.” Please forgive me for insisting that the
three letters “OFF” at the beginning of the telegram mean
“Government, official” from Paris. No French telegraph office could
transmit such a telegram if it did not come from an official authority.
This official authority is the French Delegation of the IMT in Paris,
which received the statement made by General Giraud and
transmitted it to us: “By General Giraud, French Delegation of the
IMT.”
THE PRESIDENT: Very well, the Tribunal will receive the
document under Article 21 of the Charter.
M. DUBOST: I am grateful to the Tribunal. I read further on, at
Page 150:
“On the other hand, the death of Madame Granger on 24
September 1943 is undoubtedly due to lack of care and
medicine, in spite of her reiterated requests for both. After
an autopsy of her body, which took place in the presence of
a French doctor, specially summoned from Paris after her
death, authorization was given to this doctor, Dr. Claque to
bring the four children back to France, and then to Spain,
where they would be handed over to their father. This was
refused by the Gestapo in Paris, and the children were sent
back to Germany as hostages, where their grandmother
found them only 6 months later.”
The last four lines:
“The health of Madame Giraud, her daughter Marie
Theresa, and two of her grandchildren has been gravely
impaired by the physical, and particularly by the moral,
hardships of their deportation.”
As a reprisal for the escape of General Giraud, 17 persons were
arrested, all innocent of his escape.
I have frequently shown that in their determination to impose
their reign of terror the Germans resorted to means which revolt the
conscience of decent people. Of these means one of the most
repugnant is the call for informers.
Document F-278(b), Page 152, which we submit as Exhibit
Number RF-408, is a reproduction of an ordinance of 20 December
1941, which is so obviously contrary to international law that the
Foreign Ministry of the Reich itself took cognizance of it. The
ordinance of 27 December 1941 prescribes the following:
“Whosoever may have knowledge that arms are in the
possession or keeping of an unauthorized person or
persons is obliged to declare that at the nearest police
headquarters.”
The Ministry of Foreign Affairs in Berlin, on 29 June 1942,
objected to the draft of a reply to the French note, which we do not
have here but which must have been a protest against this
ordinance of 27 December 1941. The Tribunal knows that in the
military operations which accompanied the liberation of our land
many archives disappeared, and therefore we cannot make known to
the Tribunal the protest to which the note of 29 June 1942, from the
German Foreign Ministry refers.
Paragraph 2 summarizes the arguments of the French protest.
The French evidently had written: If German territory were occupied
by the French, we would certainly consider as a man without honor
any German who denounced to the occupying power an infraction of
their laws, and this point of view was taken up and adopted by the
German Foreign Ministry. The note continues:
“As a result of consideration of this matter, the Foreign
Office considers it questionable whether punishment should
be inflicted on whomsoever fails to denounce a person
possessing or known to possess arms. Such a prescription
of penalty under this general form is, in the opinion of the
Foreign Office, the more impracticable in that it would offer
the French the possibility of calling attention to the fact that
the German Army is demanding of them acts which would
be considered Criminal if committed by German citizens.”
This German note, I repeat, comes from the Reich Ministry of
Foreign Affairs and is signed “Strack.” There is no more severe
condemnation of the German Army than that expressed by the Reich
Ministry of Foreign Affairs itself. The reply of the German Army will
be found by the Tribunal on Page 155, “Berlin. 8 December 1942.
High Command of the Wehrmacht.” The High Command of the
Wehrmacht concludes:
“. . . since it does not seem desirable to enter into
discussion with the French Government on the questions of
law evoked by them, we too consider it appropriate not to
reply to the French note.”
This note begins, moreover, by asserting that any relaxing of the
orders given would be considered as a sign of weakness in France
and in Belgium.
These are not the signs of weakness that the German Army gave
in our occupied countries of the West. The weakness manifested
itself in terror; it brought terror to reign throughout our countries,
and that in order to permit the development of the policy of
extermination of the vanquished nations which, in the minds of all
Nazi leaders, remained the principal purpose, if not the sole purpose,
of this war.
This terrorist policy, of which the Tribunal has just seen examples
in connection with the repression of attacks by our French Forces of
the Interior on the enemy, developed without any military necessity
for it in all the countries of the West. The devastations committed by
the enemy are extremely numerous. We shall limit our presentation
to the destruction of Rotterdam at a time when the city had already
capitulated and when only the question of the form of capitulation
had to be settled; and secondly, to a description of the inundations
which the German Army caused, without any military necessity of
any sort, in 1945 on the eve of its destruction when that Army
already knew that it had lost the game.
We have chosen the example of Rotterdam because it is the first
act of terrorism of the German Army in the West. We have taken the
inundations because, without her dykes, without fresh water, Holland
ceases to exist. The day her dykes are destroyed, Holland
disappears. One sees here the fulfillment of the enemy’s aim of
destruction, formulated long ago by Germany as already shown by
the citation from Hitler with which I opened my speech, an aim
which was pursued to the very last minute of Germany’s existence as
is proved by those unnecessary inundations.
We submit to the Tribunal Document F-719 as Exhibit Number
RF-409, which comprises Dutch reports on the bombing of
Rotterdam and the capitulation of the Dutch Army. On Pages 38 and
39 of the second document book are copies of the translations of
documents exchanged between the commander of the German
troops before Rotterdam and the colonel who was in command of
the Dutch troops defending the city.
Captain Backer relates the incidents of that evening which ended
with the burning of the city. At 1030 hours a German representative
appeared with an ultimatum, unsigned and without any indication of
the sender, demanding that the Dutch capitulate before 1230 hours.
This document was returned by the Dutch colonel, who asked to be
told the name and the military rank of the officer who had called
upon him to surrender.
At 1215 hours Captain Backer appeared before the German lines
and was received by a German officer. At 1235 hours he had an
interview with German officers in a dairy shop. A German general
wrote his terms for capitulation on the letter of reply, which the
representative of the Dutch General Staff had just brought to him.
At 1320 hours Captain Backer left the place, this dairy shop
where the negotiations had taken place, with the terms to which a
reply had to be given. Two German officers escorted him. These
escorting officers were protected by the flight of German aircraft,
and red rockets were fired by them at 1322 and 1325 hours. At 1330
hours the first bomb fell upon Rotterdam, which was to be
completely set on fire. The entry of the German troops was to take
place at 1850 hours, but it was put forward at 1820 hours. Later the
Germans said to Captain Backer that the purpose of the red rockets
was to prevent the bombing. However, there had been excellent
wireless communication from the ground to the aircraft. Captain
Backer expressed his surprise that this should have been done by
means of rockets.
The work on the inundation of the “Wieringermeer” polder began
on 9 and 10 April 1945. I quote a Dutch document. That day
German soldiers appeared on the polder, gave orders, and placed a
guard for the dyke.
“On 17 April 1945 at 1215 hours the dyke was dynamited
so that two parts of it were destroyed up to a height
somewhat lower than the surface of the water of the
Ijesselmeer . . . .
“As for the population, they were warned during the night
of 16 to 17 April”—that is, at the time when the water was
about to flood the polder—“In Wieringerwerf the news
received by the mayor was passed from house to house
that at noon the dyke would be destroyed. Altogether for
this great polder, with an area of 20,000 hectares, not more
than 8½ to 9 hours were granted for evacuation . . . .
Telephone communications had been completely
interrupted; and it was impossible to use automobiles,
which meant that some individuals did not receive any
warning until 8 o’clock in the morning . . . .
“The time given to the population was, therefore, too short
for the evacuation . . . .
“The looting in the flooded polder has already been
mentioned. During the morning of 17 April, on the day of
the disaster, groups of German soldiers begin to loot . . .
These soldiers came from Wieringen . . . Moreover, they
broke everything that they did not want to take . . .”
This polder by itself covers half of all the flooded lands in
Northern Holland. The polder was flooded on 17 April, when defeat
was already a fact as far as the German Army was concerned. The
Dutch people are seeking to recover the land which they have lost.
Their courage, industry and energy arouse our admiration, but it is
an immense loss which the German Army inflicted upon those
people on 17 April.
Terrorism and extermination are intimately interwoven in all
countries in the West.
 Document C-45, which we submit as Exhibit Number RF-410 and
which is the first in the document book, is an order of 10 February
1944 showing that repression, in the minds of the leaders of the
German Army, was to be carried out without consideration of any
kind:
“Fire must be immediately returned. If, as a result, innocent
people are struck, it is to be regretted but it is entirely the
fault of the terrorists.”
These lines were written over the signature of an officer of the
general staff of the German Military Command in Belgium and
Northern France. This officer was never denounced by his superiors
as can be seen by the document.
Document F-665, submitted as Exhibit Number RF-411, Page 2 of
your document book:
“The search of suspected villages requires experience. SD
or GFP (Secret Field Police) personnel should be called
upon. The real accomplices of the guerillas must be
disclosed, and apprehended with all severity. Collective
measures against the inhabitants of entire villages (this
includes the burning of villages) are to be taken only in
exceptional cases and may be ordered only by divisional
commands or by chiefs of the SS and Police.”
This document is dated 6 May 1944. It comes from the High
Command of the Wehrmacht; and it, or at least the covering letter, is
signed by Jodl.
This document involves not only the Army General Staff, but the
Labor Service—that is to say, Sauckel—and the Todt Organization—
that is to say, Speer. Indeed, in the next to the last paragraph we
may read:
“The directive . . . is applicable to all branches of the
Wehrmacht and to all organizations which exercise their
activities in occupied territories (the Reich Labor Service,
the Todt Organization, et cetera).”
 These orders, aimed at the extermination of innocent civilian
populations, were to be carried out vigorously but at the price of a
constant collusion of the German Army, the SS, the SD, and the
Sipo, which the people of all countries of the West place together in
the same horror and in the same reprobation.
In the war diary of General Von Brodowski submitted this
morning under Exhibit Number RF-405, an excerpt of which is to be
found on Pages 3, 4, and 5 of the document book, it is stated that
repressive operations were carried out:
“An action against terrorists was undertaken in the
southwestern area of the Department of Dordogne near
Lalinde, in which a company of Georgians of Field Police,
and members of the SD took part . . .”
Dated 14 June 1944 is a statement on the destruction of
Oradour-sur-Glane. I shall come back to the destruction of this
village. “600 persons are said to have been killed,” writes General
Von Brodowski. It is underscored in the text.
“The whole male population of Oradour has been shot.
Women and children took refuge in the church. The church
caught fire. Explosives had been stored in the church. Even
women and children perished.”
We shall let you know the results of the French inquiry. The Tribunal
will see to what degree General Von Brodowski lied when he
described the annihilation of Oradour in these terms.
Concerning Tulle:
“On 8 July 1944 in the evening the barracks occupied by
the 13th Company of the 95th Security Regiment were
attacked by terrorists. The struggle was terminated by the
arrival of the Panzer division, ‘Das Reich.’ 120 male
inhabitants of Tulle were hanged, and 1,000 sent to the SD
at Limoges for investigation.”
THE PRESIDENT: M. Dubost, could we see the original of this
document?
 M. DUBOST: I showed it to you this morning, Mr. President, when
I submitted it. It is rather a large document, if you will remember,
Sir.
THE PRESIDENT: Yes. We would like to see it.
DR. ROBERT SERVATIUS (Counsel for Defendant Sauckel): I
should like briefly to rectify an error now, before it is carried any
further.
The French Prosecutor mentioned that certain people were put at
the disposal of the Arbeitsdienst. I should like to point out that
Arbeitsdienst is not to be confused with the Arbeitseinsatz. The
Arbeitseinsatz was ultimately directed by Sauckel, whereas the
Arbeitsdienst had nothing whatsoever to do with Sauckel. I should
like to ask the Tribunal to take judicial notice of that distinction.
THE PRESIDENT: On account of a technical incident, the Tribunal
will adjourn.
[A recess was taken.]

THE PRESIDENT: The attorney for Sauckel, I think, was

addressing the Tribunal.
DR. SERVATIUS: I had pointed out the difference between the
Arbeitsdienst and the Arbeitseinsatz. The French prosecuting
attorney apparently confused the Arbeitsdienst with the
Arbeitseinsatz, for he said that the Arbeitsdienst was connected with
Sauckel. That is not so. The Arbeitsdienst was an organization for
premilitary training which existed before the war and in which young
people had to render labor service. These young people were to
some extent used for military purposes. The Arbeitseinsatz was
concerned solely with the recruiting of labor to be used in factories
or other places of work. It follows, therefore, that Sauckel cannot be
associated with the accusations that were made in this connection.
That is what I wanted to say.
M. DUBOST: The two German words were translated in an
identical manner in French. A verification having been made, the
remarks of the defense are correct and Sauckel is not involved, but
only the Army.
 THE PRESIDENT: Very well.
M. DUBOST: Here are a few examples of terrorist extermination
in Holland, in Belgium, and in other occupied countries of the West.
In Holland, as one example out of a thousand, there were the
massacres of Putten of 30 September 1944. They are included in
Document Number F-224, which we submit as Exhibit Number RF-
324 and which is to be found on Page 46 of the document book. On
30 September 1944 an attack was made at Putten by members of
the Dutch resistance against a German automobile. The Germans
concluded that the village was a refuge for partisans. They searched
the houses and assembled the population in the church.
A wounded German officer had been taken prisoner by the Dutch
resistance. The Germans declared that if this officer was released
within 24 hours no reprisals would be made. The officer was
released, after having received medical care from the soldiers of the
Dutch resistance who had captured him. However, in spite of the
pledge given, reprisals were made upon the village of Putten, whose
inhabitants were all innocent.
I now cite Paragraph 2 of the Dutch report:
“The population gathered in the church was informed that
the men would be deported and the women had to leave
the village because it would be destroyed.
“150 houses were burned down (the total amount of
houses in the built-up area being about 2,000).
“Eight people, amongst whom a woman who tried to
escape, were shot.
“The men were taken to the concentration camp at
Amersfoort. Amongst them were many accidental passers-
by who had been admitted into the closed village but who
had been prevented from leaving the place.
“At Amersfoort about 50 people were selected; and during
the transport, 12 jumped out of the train. 622 men were
eventually deported to Auschwitz. The majority of those
died after two months.
 “From the 622 deported men, only 32 inhabitants of the
village of Putten and 10 outsiders returned after the
liberation.”
In Belgium, we will cite only a few facts which are related in
Document Number F-685, already submitted under Exhibit Number
RF-394. This document is to be found on Page 48 in your document
book. It describes the murder of a young man who had sought
refuge in a dug-out. He was killed by the Germans who were looking
for soldiers of the Belgian secret army.
At Hervé the Germans fired on a lorry filled with young men and
killed two of them. The same day some civilians were killed by a
tank.
On Page 49, the summary executions of members of the secret
army are described. I quote:
“At Anhée, shots having been fired upon them, the
Germans crossed the river Meuse. They set fire to 58
houses and killed 13 men. At Annevoie, on the 4th, the
Germans came across the river and burned 58 houses.”
Then follows a report on destruction, useless from the military
point of view:
“. . . At Arendonck, on the 3rd, 80 men were killed, five
houses were burned. At St. Hubert, on the 6th, three men
killed and four houses burned. At Hody, on the 6th,
systematic destruction of the village, 40 houses destroyed,
16 people killed. At Marcourt, 10 people were shot, 35
houses were burned. At Neroeteren, on the 9th, 9 people
were killed. At Oost-Ham, on the 10th, 5 persons were
killed. At Balen-Neet, on the 11th, 10 persons were shot.”
Page 50 contains the description of German extortions at the
time of the temporary stabilization of the front.
“At Hechtel, the Germans having withdrawn before the
British vanguard, the inhabitants hung out flags. But fresh
German troops came to drive out the British vanguard and
 reprisals were taken; 31 people were shot; 80 houses were
burned, and general looting took place. At Helchteren 34
houses were set on fire and 10 people were killed under
similar circumstances. The same thing took place at
Herenthout . . . .
“The circumstances in which these men were executed are
always identical. The Germans search the cellars, bring the
men out, line them along the highway, and shoot them,
after having given them the order to run. In the meantime,
grenades are thrown into the cellars, wounding women and
children.”
Another example:
“At Lommel, the unexpected return of the German soldiers
found the village with flags out. Seventeen persons who
had sought refuge in a shelter were noticed by a German.
He motioned to a tank which ran against the shelter
crushing it and killing 12 people.”
In the case of Norway we shall take an example from a
document already submitted under Exhibit Number RF-323, Pages
51 and 52 of your book:
“. . . on 13 April 1940, two women 30 years of age were
shot at Ringerike. On 15 April, four civilians, of whom two
were boys of 15 and 16 years of age, were shot in Aadal.
One of those murdered was shot through the head, and
had also been bayonetted in the stomach. On 19 April four
civilians, of whom two were women and one a little boy 3
years of age, were shot at Ringsaker.
“To avenge the death of the two German policemen, who
were shot on the 26th of April 1942 at Televaag, the entire
place was laid waste. More than 90 properties with 334
buildings were totally destroyed, causing damage to
buildings and chattels (furniture and fishing outfits)
amounting to a total of 4,200,000 Kroner.”
 In this document the Tribunal will find the continuation of the
descriptions of German atrocities committed in Norway, without any
necessity of a military character, simply to maintain the reign of
terror.
In France massacres and destructions without military purpose
were extremely numerous, and all of them were closely associated.
We submit Document F-243 as Exhibit Number RF-412. The Tribunal
will find this document on Pages 178 to 193 of the document book.
It is a long list, drawn up by the French Office for Inquiry into War
Crimes, of the towns that were destroyed and looted without any
military necessity. The Tribunal will undoubtedly be enlightened by
the reading of this document. We shall give but a few examples. In
submitting this Document F-909 as Exhibit Number RF-413, we
intend to relate the conditions under which a whole section of
Marseilles was destroyed—Pages 56, 57, and 58, of your document
book.
It is estimated that about 20,000 people were evacuated. This
evacuation was ordered on 23 January. It was carried out without
warning during the night of the 23rd to the 24th. I quote:
“It is estimated that 20,000 persons were evacuated. From
Fréjus some of them were shipped by the Germans to the
concentration camp of Compiègne. . . .
“The demolition operations began on 1 February at about 9
o’clock in the morning. They were carried out by troops of
the German engineer corps. . . .
“The area destroyed is equivalent to 14 hectares: that is
approximately 1,200 buildings.”
Inquiry was made to find those who were responsible for this
destruction. After the liberation of Marseilles the German consul in
Marseilles, Von Spiegel, was interrogated. His testimony is in
Document F-908, which we submit as Exhibit Number RF-414, Page
53 of your document book. Spiegel stated:
“I know that a very short time after the evacuation of the
old port the rumor spread that this measure had been
 brought about by financial interests, but I can assure you
that in my opinion such a hypothesis is erroneous. The
order came from the higher authorities of the Reich
Government and had only two motives—the security of
troops and the danger of epidemics.”
We do not intend to give you a complete description of the
attacks committed by the Germans but merely a few examples. We
submit Document F-600 as Exhibit Number RF-415, Page 59:
“At Ohis (Aisne) a civilian wanted to give an American
soldier some cider to drink. The Germans returned. The
American soldier was taken prisoner, and M. Hennebert was
also taken away by the Germans to a spot known as the
‘Black Mountain’ in the village of Origny en Thiérache where
his body was later discovered partly hidden under a stack
of wood. The body bore the trace of two bayonet wounds
in the back.”
I submit Document F-604 as Exhibit Number RF-416, Page 61 of
the document book. A civilian was killed in his vineyard. Young men
and girls walking along the road were killed. The motive was given
as “presence of Maquis in the region.” All these victims were
completely innocent.
Document F-904, which I submit as Exhibit Number RF-417, Page
62 of your document book. At Culoz “. . . young boys were arrested
because they had run away at the sight of the Germans. . . .” They
were reported. “. . . not one of them belonged to the resistance.”
At St. Jean-de-Maurienne—Document F-906, submitted as Exhibit
Number RF-418, Page 63 of your document book:
“On 23 August the gendarmes, Chavanne and Empereur,
dressed in civilian clothes, and M. Albert Taravel were
arrested by German soldiers without legitimate reason. The
lieutenant who was in charge of the Kommandantur
promised the officer of the gendarmes to release these
three men. This German later surreptitiously ordered his
men to shoot these prisoners.
 “Mademoiselle Lucie Perraud, 21 years of age, who was a
maid at the Café Dentroux, was raped by a German soldier
of Russian origin, under threat of a pistol.”
I will not mention any more of the atrocities described in this
document.
I now come to the Vercors. This region was undeniably an
important assembly center for French Forces of the Interior.
Document F-611, which we submit as Exhibit Number RF-419,
describes the atrocities committed against the innocent population of
this region in reprisal for the presence of men of the Maquis. This
document appears in your book on Page 69 and following. In
Paragraph 3 is an enumeration of police operations in the Vercors
area.
On 15 June, in the region of St. Donat: rape and looting.
Execution at Portes-les-Valence on 8 July 1944 of 30 hostages taken
from among the political prisoners interned at Fort Montluc at Lyons.
Police raids carried out against the Maquis of the Vercors region from
21 July to 5 August 1944. Rape and looting in the region of Crest,
Saillans, and Die. Bombing by aircraft of numerous villages in the
Vercors area and in particular at La Chapelle and Vassieux-en-
Vercors; summary execution of inhabitants of these places; looting.
Execution, after summary judgment, of about a hundred young men
at St. Nazaire-en-Royans; deportation to Germany of 300 others
from this region. Murder of 50 gravely wounded persons in the
Grotto of La Luire. On 15 June 1944, attack by German troops at St.
Donat. I quote, “The Maquis had evacuated the town several days
earlier . . . 54 women or young girls from 13 to 50 years of age were
raped by the maddened soldiers.”
The Tribunal will forgive me if I avoid citing the atrocious details
which follow. Bombing of the villages of Combovin, La Baume-
Cornillanne, Ourches, et cetera:
“The losses caused by these bombings among the civilian
population are rather high, for in most cases the
inhabitants, caught by surprise, had no time to seek shelter
 . . . 2 women were raped at Crest . . . 3 women were raped
at Saillans . . . .
“A young girl of twelve, who was wounded and pinned
down between beams, awaited death for 6 long days
unable either to sit down or sleep, and without receiving
any food, and that under the eyes of the Germans who
were occupying the village.”—A medical certificate from
Doctor Nicolaides, who examined the women who were
raped in this region.
I will pass on.
I submit Document F-612 under Exhibit Number RF-420. To
terrorize the inhabitants at Trebeurden in Brittany they hanged
innocent people, and slashed the corpses to make the blood flow.
I proceed. Document F-912 is submitted as Exhibit Number RF-
421, Page 82 of your book. It is the report of the massacre of 35
Jews at St. Amand-Montrond. These men were arrested and killed
with pistol shots in the back by members of the Gestapo and of the
German Army. They were innocent of any crime.
I submit Document F-913 as Exhibit Number RF-422—Page 96, I
am quoting:
“On 8 April 1944 German soldiers of the Gestapo arrested
young André Bézillon, 18 years of age, dwelling at Oyonnax
(Ain), whose brother was in the Maquis. The body of this
young man was discovered on 11 April 1944 at Siège (Jura)
frightfully mutilated. His nose and tongue had been cut off.
There were traces of blows over his whole body and of
slashes on his legs. Four other young men were also found
at Siège at the same time as Bézillon. All of them had been
mutilated in such a manner that they could not be
identified. They bore no trace of bullets, which clearly
indicates that they died from the consequences of ill-
treatment.”
I submit Document F-614 as Exhibit Number RF-423, at Page 98
of your document book. It describes the destruction of the village of
Cerizay, (Deux-Sèvres). I quote:
“The fire did not cause any accident to persons, but the
bodies of two persons killed by German convoys and those
of two victims of the bombardment were burned.”
This village was destroyed by artillery fire; 172 buildings were
destroyed and 559 were damaged. We now submit another
document, Document F-919 as Exhibit Number RF-424, Page 103. It
concerns the murder of a young man of Tourc’h in Finistère. The
murderers compelled the mother to prepare a meal for them. Having
been fed, they had the victim disinterred. They searched and found
that the body bore a card of identity bearing the same name and
address as his mother, brothers, and sisters, who were present and
in tears. One of the soldiers, finding no excuse to explain this crime,
said dryly before going away: “He was not a terrorist! What a pity!”
and the body was buried again. Document F-616 submitted as
Exhibit Number RF-425, Page 104, concerns the report of the
operations of the German Army in the region of Nice, about 20 July
1944. I quote:
“. . . having been attacked at Presles by several groups of
Maquis in the region, by way of reprisal, this Mongolian
detachment, as usual commanded by the SS, went to a
farm where two French members of the resistance had
been hidden. Being unable to take them prisoners, these
soldiers then arrested the proprietors of that farm (the
husband and wife), and after subjecting them to numerous
atrocities, rape, et cetera, they shot them with submachine
guns. Then they took the son of these victims, who was
only 3 years of age; and, after having tortured him
frightfully, they crucified him on the gate of the farmhouse.”
We submit Document F-914 as Exhibit Number RF-426, Page 107 of
your document book. This is a long recital of murders committed
without any cause whatever by the German Army in Rue Tronchet at
Lyons. I now read:
 “Without preliminary warning, without any effort having
been made to verify the exact character of the situation
and, if necessary, to seize those responsible for the act, the
soldiers opened fire. A certain number of civilians, men,
women, and children fell. Others who were untouched or
only slightly wounded fled in haste.”
The Tribunal will find the official report that was drawn up on the
occasion of these murders.
We submit without quoting, asking the Tribunal to take judicial
notice of it only, the report relating to the crimes of the German
Army committed in the region of Loches (Indre-et-Loire), Document
F-617, submitted as Exhibit Number RF-427, Page 115 of your
document book.
Document F-607, submitted as Exhibit Number RF-428, which is
on Page 119 of your document book, describes the looting, rape,
and burnings at Saillans during the months of July and of August
1944. I quote, “During their sojourn in the region”—referring to
German soldiers—“rapes were committed against three women in
that area.” I pass on. Document F-608, Page 120 of your document
book, submitted as Exhibit Number RF-429: A person was burned
alive at Puisots by a punitive expedition. This person was innocent.
I submit Document F-610 as Exhibit Number RF-430, Page 122 of
your document book. The whole region of Vassieux in the Vercors
was devastated. This document, Number F-610, is a report by the
Red Cross prepared prior to the liberation. I am quoting:
“We found on a farm a wounded man, who had been hit by
8 bullets in the following circumstances. The Germans
forced him to set fire to his own house, and tried to prevent
him from escaping the flames by shooting at him with their
pistols. In spite of his wounds, he was able miraculously to
escape.”
We submit Document F-618 as Exhibit Number RF-431, Page 124
of the document book. I quote, concerning people who were
executed:
 “Before being shot these people were tortured. One of
them, M. Francis Duperrier, had a broken arm and his face
was completely mutilated. Another, M. Feroud-Plattet, had
been completely disembowelled with a piece of sharp
wood. His jaw bone was also crushed.”
We submit Document 605 as Exhibit Number RF-432, Page 126.
This document describes the burning of the hamlet of des Plaines
near Moutiers, in Savoy: “Two women, Madame Romanet, a widow,
72 years old, and her daughter, age 41, were burned to death in a
small room of their dwelling, where they had sought refuge. In the
same place a man, M. Charvaz, who had had his thigh shattered by
a bullet, was also found burned.”
We now submit as Exhibit Number RF-433 the French Document
F-298, Page 127 and following in your document book, which
describes the destruction of Maillé in the department of Indre-et-
Loire. That area was entirely destroyed on 25 August 1944, and a
large number of its inhabitants were killed or seriously wounded.
This destruction and these crimes had no terrorist action, no action
by the French Forces of the Interior as a motive.
Document F-907 submitted as Exhibit Number RF-434—Page 132
and following in your document book—relates the incidents leading
to German crimes at Montpezat-de-Quercy. This is a letter written to
the French Delegation by the Bishop of Montauban, Monseigneur
Théas, on 11 December 1945. This document really explains
Document F-673, already submitted as Exhibit Number RF-392, from
which I will read. The first part consists of a letter by the French
Armistice Commission, and has been taken from the archives of the
Armistice Commission in Wiesbaden:
“On the night of 6 to 7 June last, in the course of an
operation in the region of Montpezat-de-Quercy, German
troops set fire to four farmhouses which formed the hamlet
called ‘Perches.’ Three men, two women, and two children,
14 and 4 years old, were burned alive. Two women and a
child of ten who disappeared probably suffered the same
fate.
“On Saturday, 10 June, having been fired at by two
recalcitrants at the village of Marsoulas, German troops
killed these two men. Moreover, they massacred without
any explanation all the other inhabitants of the village that
they could lay their hands on.
“Thus 7 men, 6 women, and 14 children were killed, most
of them still in their beds at the early hour when this
happened.
“On 10 June, at about 1900 hours, five Luftwaffe aircraft
attacked the town of Tarbes for half an hour with bombs
and machine guns. Several buildings were destroyed,
among them the Hôtel des Ponts et Chaussées, and the
Academic Inspectorate. There were 7 dead and about 10
wounded who were hit by chance among the population of
the town. On this occasion the general in command of the
VS-659 at Tarbes immediately informed the Prefect of the
Department of Basses-Pyrénées that the operation had
been neither caused nor ordered by him.
“Following each of these events the Regional Prefect of
Toulouse addressed to the general commanding the HVS-
564 letters in which in dignified and measured terms he
protested against the acts in question, through which
innocent women and children were deliberately killed. He
asserted very rightly that under no circumstances could
children in the cradle be considered as accomplices of the
terrorists. He requested finally that instructions be given to
avoid the recurrence of such painful events.
“Replying on 19 June to the three letters of the Regional
Prefect of Toulouse, the chief of staff of the general
commanding the head liaison staff 564 announced the
principles which determined the position taken by his chief,
which justified the acts of reprisal quoted on the following
grounds:
 “The duty of the French population is not only to flee from
terrorists but also to render their operations impossible,
which will avoid any reprisals being taken against innocent
people. In the struggle against terrorism the German Army
must and will employ all means at its disposal, even
methods of combat new to Western Europe.
“The terror raids of the Anglo-Americans also massacre
thousands and thousands of German children. There, too,
innocent blood is being shed through the action of the
enemy, whose support of terrorism is forcing the German
soldier to use his arms in the South of France.
“I beg to ask you”—concluded General Bridoux, writing to
the German Commission—“whether the French Government
is to consider the arguments cited above as reflecting
accurately the position taken by the German High
Command, in view of the facts disclosed in the first part of
the present letter.”
We now submit Document E-190 as Exhibit Number RF-435,
Page 141 of the document book, which describes the crimes
committed at Ascq by a German unit which, in reprisal for the
destruction of the railway, massacred 77 men of all categories and
all ages, among whom were 22 employees of the French State
railway, some industrialists, business men, and workmen. I quote:
“The oldest of these victims, M. Briet, retired, was 74 years
old; he was born on 3 October 1869 at Ascq. The youngest,
Jean Roques, student and son of the postmaster, was 15
years old, born on 4 January 1929 at Saint Quentin. Father
Gilleron, a priest at Ascq, and his two protegées, M. Averlon
and his son, who had fled from the coast, were also shot.”
This massacre was the cause of a protest made by the French
Government at that time, to which Commander-in-Chief Von
Rundstedt replied on 3 May 1944 (Document F-673, already
submitted as Exhibit Number RF-392, Page 154):
 “The population of Ascq bears the responsibility for the
consequences of its treacherous conduct, which I can only
severely condemn.”
General Bérard, president of the French delegation attached to
the German Armistice Commission, was not satisfied with the reply
given by Rundstedt; and on 21 June 1944 he reiterated the French
protest, addressing it this time to General Vogl, president of the
German Armistice Commission. This is still Document F-673, Exhibit
Number RF-392. I quote:
“In all, from 10 October 1943 to 1st May 1944, more than
1,200 persons were made the victims of these measures of
repression. . . .
“These measures of repression strike the innocent and
cause terror to reign among the French population . . . .
“A great number of the acts that have been mentioned took
place in the course of repressive operations directed against
population accused of having relations with the Maquis. In
these operations there was never any care taken to
discover whether the people suspected of having served
the Maquis were really guilty; and still less in this case, to
ascertain whether these people had acted voluntarily or
under duress. The number of innocent people executed is
therefore considerable. . . .
“The repressive operation in Dordogne, from 26 March to 3
April 1944, and particularly the tragic affair of Ascq, which
have already brought about the intervention of the head of
the French Government, are grievous examples. At Ascq,
especially, 86 innocent people paid with their lives for an
attempted attack which, according to my information, did
not cause the death of a single German soldier. . . .
“Such acts can only stimulate the spirit of revolt in the
adversaries of Germany, who finally are the only ones to
benefit.”
 The reply of the Armistice Commission, Document F-707,
submitted as Exhibit Number RF-436, is the rejection of General
Bérard’s request. The document is before you. I do not think it is
necessary for me to read it.
The general, on 3 August 1944, reiterated his protest. This is
Document F-673, Exhibit Number RF-392, already submitted. At the
end of his protest he writes:
“An enemy who surrenders must not be killed even though
he is a franc-tireur or a spy. The latter will receive just
punishment through the courts.”
But this is only the text of stipulations to be applied within Germany.
We submit Document F-706, Exhibit Number RF-437, which is a
note from the French Secretary of State for Defense to the German
general protesting against the measures of destruction taken by the
German troops in Chaudebonne and Chaveroche. We shall not read
this document. The Tribunal may take judicial notice of it, if it deems
it necessary.
We now come to the statement of the events of Tulle, in which
120 Frenchmen were hanged, Page 169 (Document F-673, Exhibit
RF-392). I am quoting:
“On 7 June a large group of francs-tireurs attacked the
French forces employed in the maintenance of order and
succeeded in seizing the greater part of the town of Tulle
after a struggle which lasted until dawn. . . .
“The same day, at about 2000 hours, important German
armored forces came to the assistance of the garrison and
penetrated into the city from which the terrorists withdrew
in haste. . . .”
These troops, which re-took Tulle, decided to carry out reprisals.
The French Forces of the Interior that had taken the town had
withdrawn. The Germans had taken no prisoners. The reprisals were
carried out upon civilians. Without discrimination they were arrested.
 “The victims were selected without any inquiry, without
even any questioning, haphazardly; workmen, students,
professors, industrialists. There were even among them
some militia sympathizers and candidates for the Waffen
SS. The 120 corpses which were hanged from the balconies
and lamp-posts of the Avenue de la Gare, along a distance
of 500 meters, were a horrible spectacle that will remain in
the memory of the unfortunate population of Tulle for a
long time.”
We now come to the crowning event in these German atrocities:
the destruction of Oradour-sur-Glane, in the month of June 1944.
The Tribunal will accept, we hope, the presentation of Document F-
236, which now becomes Exhibit Number RF-438. This is an official
book, published by the French Government, which gives a full
description of the events. I will give you a brief analysis of the report
which the de facto government of the time sent to the German
general who was Commander-in-Chief for the regions of the West:
“On Saturday, 10 June, a detachment of SS belonging very
likely to the ‘Das Reich’ division which was present in the
area, burst into the village, after having surrounded it
entirely, and ordered the population to gather in the central
square. It was then announced that it had been reported
that explosives had been hidden in the village and that a
search and the checking of identity were about to take
place. The men were asked to make four or five groups,
each of which was locked into a barn. The women and
children were taken to the church and locked in. It was
about 1400 hours. A little later machine-gunning began and
the whole village was set on fire, as well as the surrounding
farms. The houses were set on fire one by one. The
operation lasted undoubtedly several hours, in view of the
extent of the locality.
“In the meantime the women and the children were in
anguish as they heard the sound of the fires and of the
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