echelon or staircase-shaped forms. These echelon forms will be studied in the next
n.
section. They will also be used for m x n systems, where m
-=I-
EXERCISES
1. Use back substitution to solve each of the following systems of equations.
+
9XS
4 2X2
1 2+
-3X2
X3
X2
X4
=- - 582X3
+
-5X3
-+
(a) Xl 52X4
X4
XS
X3
2X4
2xs
2X2 ++-X3
3X2
(c)
4X3
2X2
+ -1X4
++(b)2X4
3X3=
=
2X2
4X4 - - 6
2xs
3xs
02
-
2. Write out the coefficient matrix for each of the systems in Exercise l.
3. In each of the following systems, interpret each equation as a line in the plane. For each
system, graph the lines and determine geometrically the number of solutions.
3+
X2
(a) XlXl
X2
2Xl
4X2
+X2
- ==
Xl
+
2X2
=44 = 4
(d)
(b)
62
-Xl
+
3X2
=
-2XI
Xl -
3
[i;
4. Write an augmented matrix for each of the systems in Exercise 3.
1
1
5.
-1
-2
23
4-2
-3
1 -2
-5
6 63
Write out the system (d)
of equations
1 that corresponds to each of the following augmented
(b)
3 matrices.
~
-~ I ~I ~]
]
(a)
[
-~]
i]
[!
I
2:XI
I
e two
2:
=3X2
XI2X2X2
6. Solve
4xI
+
each
=X25of
3~X3
4- =
following
systems.
+- the3X3
TO
-1
2X3
23X2 --+
++ SX3
(c)
X2
= +8
(b) 2x]XI
++++2X2X3
3xI ++ 3X2
3xIX3X2
2X2X3
(a) 5X3
2
12_
4XI
6xI
5X2 + 5X3 =2xI-3 +
(g) ~XI
+ 4X2
Xl ++
2X2 +
systems
-XI
+
3X3
2XI + 2X2
4x]
3X2 = 6
(d)
(e) 2xI +
(f)
+
3XI
X2
3
2
I
3
7
o
+
-2xI
2XI
-1
+ 2X3
+ X3 + X4
4X4
+ 3X3
+ X3 + 2X4
+ X3 + 3X4
X2
(h)
X2
3XI
XI
2XI
III
2
X3
X2
+
+
X2
3X2
o
7
6
6
i
4XI
+
3X2
=
+
+
2xI
and
5
4xI
-1
X2
3X2
have the same coefficient matrix but different right-hand sides. Solve both systems
simultaneously by eliminating the first entry in the second row of the augmented matrix
[24
31 [35
-1]1
and then performing back substitutions for each of the columns corresponding to the
right-hand sides.
8. Solve the two systems
+
+
+
XI
2xI
Xl
2X2
5X2
3X2
+
+
2X3
1
XI
X3
9
2xI
+
+
4X3
=9
XI
+
2X2
2X3
9
9
5X2
+
X3
3X2
+
4X3
= -2
by doing elimination on a 3 x 5 augmented matrix and then performing two back substitutions.
9. Given a system of the form
-mlxI
-m2xI
where ml,
m2,
+
+
X2
X2
=
=
bl
b2
bI, and b2 are constants:
(a) Show that the system will have a unique solution if ml
(b) If ml
=
=1=
m2·
m2, show that the system will be consistent only if
h = b2·
(c) Give a geometric interpretation to parts (a) and (b).
10.
Consider a system of the form
allXI
+
aJ2X2
a21xI
+
a22x2
= 0
= 0
~
0
augmented
The reduced row echelon form of the augmented matrix for this system is
[~
1 -1 0
o
-i0
00]
There is one free variable X3. Setting X3 = 3, we obtain the solution (5,3,3),
and the general solution consists of all multiples of (5,3,3). It follows that
the variables Xl, X2, X3 should be assigned values in the ratio
Xl : X2 : X3
=5 :3 :3
This simple system is an example of the closed Leontief input-output
model. Leontief's models are fundamental to our understanding of economic
systems. Modem applications would involve thousands of industries and lead
to very large linear systems. The Leontief models will be studied in greater
detail later in Section 8 of Chapter 6.
i
20I1 4-13
30-1
2011 24 023-2
I 0(f)3 (f)00 0 001 0I 0
EXERCISES
-solution,
echelon
form?
(e)
it.1. ~Which
~
] ; find
of the following
in row echelon
form?has
Which
are in reduced
indicate
whether
the corresponding
linearmatrices
system isare
consistent.
If the system
a unique
(b)
matrix
is
in
row
echelon
n
!]
-i]
~ ](h)
:]
-2
g[!(b)
[[g(oj
1
i]
-;
]
(eJ
[!
[[!
g[g form. For each case,
(0)
[g
[~
(a) [~
(,)
1
!]I I
n
~]
row
[g
3. In each of the following, the augmented matrix is in reduced row echelon form. In each
case, find the solution set to the corresponding linear system.
(a)
0 1
[ 0013
1 0
0
0
5
-2]
(b)
0 0
[100014
1 3
0 2]
0
(f)
4Xl
2+
3X2
33X2
1=
X3
x2
4X3
=
6
2X3
3X3
=
4X4
1X4
01
01
5X4
04X2
X4
53X2
=
-(f)
0X3
05+
-30
-37x3
+
26 0-2
X3
2Xl
X2
X3
=
+
X3
02-1
+
+
=
-X4
=+
5and
0X4
(b)
8X2
2X3
-Xl
--X4
+
3X2
X3
+
-1
2X4
=
4X2
=
Xl
Xl
-+
-2XI
+
+
2X2
3X2
+2X3
x3
-=
=
=
+32x]
=
+517
- (b)
X4++
=
3X2
4X34 =
= 7
-Xl
2Xl+
Xl
(1)
2Xl
-+
2X2
=Matrices
9
-2XI
3X3
=4X2
+X3
14
3X2
(d)
+=2X2
-4XI
6X2
=+2X2
88X2++ =2X3
Ilxl
Xl
-5XI
2X2
-8
=-X3
3+
3Xl
(h)
-02+
(j)
Chapter
1
Systems
of7Xl
Equations
02X2
~I~]
-!]
0
of the
free
free
variables,
transform
it
reduced
row
echelon
form
and
allform.
solutions.
backlist
equivalent
substitution
system
tovariables.
find
whose
theto
coefficient
row
echelon
Indicate
If is
thein
system
isfind
consistent
and
therewhether
are
the
system
is consistent.
If
theunique
systemsolution.
is matrix
consistent
and
involves
no
free
variables,
use
(i)
5. For
each of the following
(d) [~ systems of equations, use Gaussian elimination to obtain an
(a) (a)
-~ ]systems.
each of the following
make a list of the lead variables[ ~and a second
=2x] 4
[~
il[
~
15.
Let
(CI, C2)
be a solution to the 2 x 2 system
allXI
~
a21XI
Show that for any real number
16.
a
+
+
o
a12X2
= 0
anX2
the ordered pair (acI,
aC2)
is also a solution.
In Application 3 the solution (6, 6, 6, 1) was obtained by setting the free variable
X4
= 1.
(a) Determine the solution corresponding to X4 = O. What information, if any, does this
solution give about the chemical reaction? Is the term "trivial solution" appropriate
in this case?
(b) Choose some other values of X4, such as 2, 4, or 5, and determine the corresponding
solutions. How are these nontrivial solutions related?
1 7. Liquid benzene burns in the atmosphere. If a cold object is placed directly over the
benzene, water will condense on the object and a deposit of soot (carbon) will also
form on the object. The chemical equation for this reaction is of the form
Xl
Determine values of
Xl,
C6H6
X2, X3,
+ X202
and
X4
~
X3C
+ X4H20
to balance the equation.
18.
Nitric acid is prepared commercially by a series of three chemical reactions. In the first
reaction, nitrogen (N2) is combined with hydrogen (H2) to form ammonia (NH3). Next
the ammonia is combined with oxygen (02) to form nitrogen dioxide (N02) and water.
Finally, the N02 reacts with some of the water to form nitric acid (HN03) and nitrogen
oxide (NO). The amounts of each component of these reactions are measured in moles
(a standard unit of measurement for chemical reactions). How many moles of nitrogen,
hydrogen, and oxygen are necessary in order to produce 8 moles of nitric acid?
19.
In Application 4, determine the relative values of
goods is as described in the following table.
11
6F
:3
22:
:
M
:3
C
3
Xl,
X2,
and
X3
1
11
ell
Mil
F
1
I
20.
Determine the amount of each current for the following networks.
3 ohms
il~
, ~BII 2ohm~
16 volts
(b)
i2
A
(a)
"
.. A..M
il
B
12
i3
ms
2 ohms 20ohms
2 ohms
rOI'" h
if the distribution of
acquire or update as many as 10 million Web pages in a single day. Although
the database matrix for pages on the Internet is extremely large, searches can
be simplified dramatically since the matrices and search vectors are sparse;
that is, most of the entries in any column are O's.
For Internet searches, the better search engines will do simple matching
searches to find all pages matching the key words, but they will not order
them on the basis of the relative frequency of the key words. Because of the
commercial nature of the Internet, people that want to sell products may deliberately make repeated use of key words to ensure that their Web site is higWy
ranked in any relative frequency search. In fact, it is easy to surreptitiously
list a key word hundreds of times. If the font color of the word matches the
background color of the page, then the viewer will not be aware that the word
is listed repeatedly.
For Web searches a more sophisticated algorithm is necessary for ranking
the pages that contain all of the key search words. In Chapter 6 we will study
a special type of matrix model for assigning probabilities in certain random
processes. This type of model is referred to as a Markov process or a Markov
chain. In Section 3 of Chapter 6 we will see how to use Markov chains to
model Web surfing and obtain rankings of Web pages.
REFERENCES
1. Berry, Michael W., and Murray Browne, Understanding Search EnginesMathematical Modeling and Text Retrieval, SIAM, Philadelphia, 1999.
EXERCISES
1. If
A
-2 0 1
=
B
and
3122 1 4]
[
= -3
[
1
21 -40
compute:
(a) 2A
(e) AB
+B
(b) A
(f) BA
(c) 2A - 3B
(g) ATBT
n
(d) (2A)T
(h) (BA)T
-
(3Bl
2. For each of the following pairs of matrices, determine whether it is possible to multiply
the first matrix times the second. If it is possible, perform the multiplication.
(a)
(c)
iI
-2
[35
0
0 2I) [2~ 1
1
[ 00245
1 43]
4
1
[3
(b)
6 -4
[1
[48 -6
-2]
(d)
[i
1
2]
n [~
2 3)
~)
(e) [42
61
~] [~
~
]
[-! I [3
(0
2 4 5)
3.
For which of the pairs in Exercise 2 is it possible to multiply
the first, and what would the dimension of the product be?
4.
Write each of the following
systems of equations
the second matrix times
as a matrix equation.
++
+ ++
= 3A + 2A
X3
(c)-=
2X3
2+0(b)
46Xl2X2
(a)Xl X2
3Xl
X2 =- 5
X2
==2Xl
I (b)(c)3Xl
X3
6A(AT)T
=
- 3(2A)
2X2
2Xl
= A-
+
2X3
= 7
(a)
SA
~]
[1
A~
verify
verifythatthat
-
2
-2
7.
I
6.
If
(c)
(a)
If
~(A
]+ B)T = AT
A + B = B +A
3
3(AB)
=
+
BT
(3A)B = A(3B)
A _ [4
and
i]
A~
B
[
23 -40 ]
= [ _~
;
(b)
3(A
+ B) = 3A + 3B
and
(b)
=
B
(AB)T
=
[i :]
BTAT
8. If
3 '
A= [21 4]
B
= [ -20 41] '
C=[~
~]
verify that
(a)
(c)
9.
(A
A(B
+ B) + C = A + (B + C)
(b)
(d)
+ C) = AB + AC
Prove the associative
law of multiplication
A = [alla21
and show that
an
a12],
B
=
(AB)C
)
I
(AB)C
(A
= A(BC)
+ B)C =
AC
for 2 x 2 matrices;
b21
[bll
=
bn
h2],
A(BC)
C
+ BC
that is, let
=
C21
[Cll
Cn
Cl2]
00
59
Exercises
10.
Let
11.
and
A2
What will
A3 .
[-1 -i)
=
A
Compute
An
turn out to be?
Let
1
1
1
1
1
2: -2: -2:
1
A
I -2:1
=
1
1
1
2: -2: -2:
Compute
A3.
What will
A2n
and
A2n+l
1
2:
turn out to be?
I
12. 0 Let 0
13.
and
A2
1
2:
-2:1 -2:1
1
-2:
2:-2:
-2:
I
1
-2:
!]
Given
A- [~
A = [~
_;),
b= [6 ) ,
c = [ =; )
(a) Write b as a linear combination of the column vectors 31 and 32.
(b) Use the result from part (a) to determine a solution to the linear system Ax
Does the system have any other solutions? Explain.
(c) Write c as a linear combination of the column vectors 31 and 32.
=
b.
A and b, determine whether the system Ax = b
is consistent by examining how b relates to the column vectors of A. Explain your
answers in each case.
2
(a) A = [ _; ) ,
(b) A = [~ i),
b = [; )
b = [ i)
14. For each of the following choices of
2
2
-~
n
[i
Ul
b~
Show that if d
=
alla22
- a21al2
A-I
A
=
i=-
0, then
an
al2 )
[alla21
d [ -a21
=~
an
)
16.
Let
17.
Prove that if A is nonsingular then A T is nonsingular and
A
be a nonsingular matrix. Show that
all
-al2
(AT)-1
[Hint:
18.
Let
..,.
-'-
A
'lI7
(AB)T
=
A
=
-1 is also nonsingular and
i-ha
-1 )-1
=
A.
(A-l)T
BTAT.]
be an n x n matrix and let x and y be vectors in
i-han
(A
.•...........•
t .•..
;-v
A
.•...•...•
n •..i-
h""
n-n 1.....•..
C1; .•..••
Rn.
Show that if
Ax
= Ay and