1.2 Matrices
19
T hen
AT
=
H~l
- 2
C
T
=
[~
-3
2
-n
U n
3
BT =
- I
2
Hl
DT =
and
[2
£T =
- I
3]'
•
Key Terms
Matrix
EqU3J mmrices
Rows
/l-vector (or vector)
Columns
Rn , c n
Size of a matrix
Square matrix
Main diagonal
Element or entry of a matrix
0, zero vector
GoogJe
Matrix addition
Scalar multiple
NU
Difference of matrices
Summation notation
Index of summation
Linear combination
Coefficients
Transpose
Exercises
l. LeI
A =
[~
(h j
A
~ [~
0
0
and
0
I
I
0
0
0
0
0
0
4. If
c~
["c -+b
d
(a) Whatisa I2.an.(I2l'!
~]
<+d]
(/ - b
=
n
[4
[0
find a, b, (". and d.
(b) Whatish ll .b31 '!
5. I f
[a + 2b
(e ) Whatis c 13.C31 .(:31 ?
2c + d
2. Determine the incidence matrix associated with each of
the following graphs:
-2]
2a - bH4
c - 2d
-3 .
4
find a, b, c . and d.
(. j
(hj
~'
III
£xercise~'
6 rhrvugh 9, lei
P,
PI
[~
2
~ [~
- I
p~
P,
A =
3. For each of the following incidence matrices. construct a
graph. Label the vertices P I. 1'2 . . ... Ps.
c
E
~ [~
-4
I
2
:1 B~ [i
lJ
J
D =
F
~
n
[~
-2]4 .
[-4 :]
2
20
Chapter 1 Linear Equations and Matrices
",ul
0
~ [~ ~ ~]
17. Show lhat the summation notation satisfies the following
properties:
(a)
6. If possible. compute the indicated linear combination:
(e)
D- F
;=1
(b) A+B
(a) C + £and£+C
(bl
(d ) -3C - 50
(I) 2B + F
(e) 2C - 3£
(b ) 312A) and 6A
+ 2A and 5A
2(D + F) and 2D +
2F
3( B
+ D)
8. Ifpossible. compute the following:
(a) A T and (AT)T
(b)
+ £)T and CT + £T
+ 3fY
(d ) D _ D T
2AT + B
(I) (3 D - 2F)T
(b) (C
(e) (2D
(e)
(e)
(e) (3 B T
(b) (A _ B )T
2A) T
_
(d) (3 A T
(e) (_ A )T and _ (A T)
. [' 0].
10. Is the matnx
0
2
(I) (C
_
5B T
,r
+ E + F T)T
..
.
.
Inces
[' '].
0
- 3
. . of the maa hnear comblllatlOn
[, O] [' 0] .
0
1 and
0
0 ? Justify your answer.
12. Let
2
2
2
If). is a real number. compute AlJ - A.
o
,
°
13. If A is an /I x /I matrix. what are the entries on the main
diagonal of A - AT? Justify )·our answer.
14. Explain why every incidence matrix A associated with a
graph is the same as A T.
15. Let the /I x /I matrix A be equal to A T. Briefly describe
lhe pallern of the entries in A.
16. If x is an /I-vector. show that x + 0 = x.
~ (t",b } [t u'][~bil
21. A brokerage finn records the high and low values of the
price of IBM slock each day. The infonnation for a given
week is presented in two vectors. I and b. in RS. showing
the high and low values. respecti vely. What expression
gives the avemge dai ly values of the price of IBM stock
for the entire 5-day week?
a lmear combmatlOn of the matn -
1 and ['0 OJ
0 ? Ju,tl.fy your answer.
OJ
.
II. Is the matnx
t (~ ,) ~ ' '
20. A large steel manufacturer. who has 2000 employees.
lists each employee·s salary as a component of a \·ector
u in R2000. If an 8% across-the-board salary increase has
been approved. find an expression in vo lving u that gives
all the new salaries.
9. If possible. compute the following:
(a) (2A) T
t e""")~c (t,,,,,)
(u)t('H' I ~(tu.) +u
(e) (2 +3) Dand2D+3D
(I)
;=1
19. Identify the following expressions as true or false. If true.
prove the result: if faIse. give a counterexample.
(e) 3A
(d )
;=1
18. ShOWlh"t(~U'I)~~(tu'i )
7. If possible. compute the indicated linear combination:
(a) 3D+2F
L" (r, + .~; )(/; = L"
r;(/,"
+ L Si(/i
.I.
22. For the software you are using. determine the commands
to enter a ma tri x. add matrices. multi ply a scalar times a
matrix. and obtain the transpose of a matrix for matrices
with numerical entries. Praclice the commands. using the
linear combinations in Examp le 13 .
.1. . 23. Determine whether the software you are llsing includes
a computer algebm system (C AS ). and if it does. do the
following:
(a ) Find the command for entering a symbolic matrix.
Ollis command may be different th an thaI for entering a numeric matrix.)
(b) Enter seveml symbolic matrices like
A
~
[,
u
~]
and
B~
u
[d
b
;1
Compute expressions like A + B. 2A. 3A + B.
A - 2 B. A T + BT. etc. (In some systems you must
30
Chapter 1 Linear Equations and Matrices
Writing Ax = b as a linear combination of the columns of A as in (6), we have
•
The expression for the linear system Ax = b as shown in (6), provides an
important way to think about solutions of linear systems.
Ax = b is consistent if and only if b can be expressed as a linear combination
of the columns of the matrix A.
We encounter this approach in Chapter 2.
Key Terms
Dot product (inner product)
Matrix~vector product
.i'
Exercises
III £Hm:isel' I alld 2, (;Ompllle
I. (a)
Coefficient matrix
Augmented matrix
a· b.
, ~[a b ~[ _~]
[_;]
(b )
,~ [ =;J b ~
(cJ
,~ ul b ~m
Idl
3.
L" '~ b ~[ -;llr'. b ~17.fiOd'.
4. Determine the value of x so that v· w = O. where
5. Detennine values of x and y so that v • w = 0 and v· u =
Owb<re ' ~
, ~m b ~m
[;]
" ~Hl,"d U ~ m
6. Determine values of.{ and y so that v • w = 0 and v · u =
O.wb,re,~ [;] w ~ [-n,"d U~
2. (a)
,~ [ ;J b ~m
(b )
'~ [ _ :l b ~[:]
7.
(c)
, ~m b ~n]
8. F;od ,II ,,'," of, '"
,~ [H b ~m
9. F;od,1I
(d )
[-H
Let w= [;~:: lcomputew. w.
' '0<
U·U~
50. wh,,,
"',,,of, ,",b" ,., ~ ,. wh", ,
U~ [!1
~ [-1].
1.3
[~
10, lelA =
Compu te th e following entries of A8 :
CO/u;der t"e/ollowing matrices/or
2 3]
~ [;
A
I
4
£\'l' l t';se,Y
II t"mugh 15:
-I
.
E
(b) the (2. 3) entry
C<) Ihe (3, I ) entry
Cd) the (3. 3) c ntry
[I
n
~]andD=
[-;
A
-I]
= [~
4 .
Show Ihat A8/= BA.
(11111
,,-=[-I~ ~2] .
20. If A is Ihe matrix in Example 4 :md 0 is Ihe 3 x 2 matrix
e"cry onc of whose entries is zero, compute AD.
111 £urc:i,\'e,\' 2J (/1Il122 , 11'1
11. If possible, co mpute the following:
(a) A B
(b) BA
(d) C 8 + 0
(e) A8 + D 1. whcre 0
- I
2
(e) F T[
1
-2
= OO
12. If possible, compUle the following:
+8
(b) £C
Cd) £8 + F
(e) FC
(a) OA
(e) CE
all(/
o
+ f)
3
2
13, If possible, compute the following:
(a) FD - 3B
[e)
F T8
( b) AB - 2D
+0
(d) 2F - 3( A£)
(e) 80 + A£
14. If
po~ib l e.
A (C+ E )
(e) (2A8 )T and 2( A8 )T
~]
- I
-3
5
2 1. Using the methexl in Example 11. co mpute the following
columns of A8 :
(a) thc first column
compute the following:
(a) A( BD)
(e)
-~l compute OIl
u,
D- [3
2 -2]5 .
n
I
~ [~ 2
-4
(a) Ihe (1. 2) entry
18. [I' f -, = 0
and ' l l> .
19.
c ~ [;
31
17. "'A ~ H n,"dB ~ [: -; :j
2
-I
[~]'findXand Y,
IfA8 =
Matrix Multiplication
(b) the third column
22. Using the methexl in Example II . compute the following
( b ) ( AB ) I>
co lumn.~o f
(d ) AC + A£
(a) the second column
( 0 A(C - 3 £ )
AB :
(b) the fourth column
23, Let
IS. If possible, compute the followin g:
(a) AT
(b l ( AT)T
(e) (A8) T
(d) B1' A T
[e) (C
(I)
+ £)1' 8
and C T B
+ £ TB
Express A c as a linear combination of the columns of A.
A (28 ) and 2( A8 )
16. Le I A = [1
C = [ -3
2
0
(a) AB T
Cd ) A T 8
(g) 8 /' CAA T
24. Lot
-3 ],8 = [ - 1 4
2].and
I]. If possible. compute the following :
(b) CA T
(e) CC T
(e) (HA T) C
(0 CTC
-2
4
o
-;]
-2
8 =
[I -I]
~
~ .
Express th e columns of A 8 as linear combinations of the
columns of A.
32
25.
Chapter 1 Linear Equations and Matrices
LetA=[~
:l,"dB~
-3
2
m
(, ) Verify lhal A B = 331 + 5a 2 + 2 3 ) , where
jlh column of A for j = 1.2.3.
33. Write the following linear system in matrix fonn:
hi
3j
3x 2
is the
hi -
26. (, ) Find a value of r so that A 8 T = 0, where
A = [r 1 - 2] and B = [I 3
(, )
- IJ.
(b) Gille an alternative way to write this product.
27. Find a value of r and a \lalue of ~' so that A S T = D. where
A=[l r l]andB=[ -2 2 .r]'
28. (a ) LeI A be an III x II matrix with a row consisting entirely of zeros. Show Ihal if B is an II x p matrix.
(hen A B has a row of zeros.
(b ) LeI A be an II! x f! m:Jlrix with a column consisting
entirely of zeros and let B be p x m. Show Ihat BA
has a column of zeros.
~ ~]With aj =lhejthcolumnOfA.
,r"
[
a~ a l
af3 2
ai 3 2
33 3 1
aj 3 2
,
(b )
U
1
- I
[-~
= 0
- I
-,
0
,
3
-,
0
1
0
0
0
3
n
n
35. How are the linear systems obtained in Exercise 34 related?
36. Write each of the following linear systems as a linear
combination of the columns of the coefficient matrix:
(a)
3xI + 21"1 + Xl 4
XI - _11+4x; = -2
+ Xz = 3
2xI - _I! = - 2
3xI + _I! =
I
(b) - XI
j = L 2, 3. Verify IlwI
A TA=
X2
34. Write the linear system whose augmented matri x is
.
(b) Venfyth:llA
B = [(roW1(A))B]
(
()
.
row: A) B
29. LeIA=[ - !
+ 112 = 0
+ x] =O
37. Write each of the following linear combinations of
columns as a linear system of the form in (4):
30. Consider the followinJ.! linear system:
+ 3X2
211
-
3X1
+
X~
+ 2X1
3_l 1
211+3x2
l)
+ X5 =
+ 3xs =
4X4
+
X4
+
_IS
7
- 2
3
38. Write each of the fo llowing as a linear system in ma trix
fonn:
5.
=
(a) Find the coefficient matrix.
(b) Write the linear system in matrix form.
(c) Find the augmented matrix.
31. Write the linear system whose augmented matrix is
[
-2
-3
- I
2
o
7
4
8
1
o
o
o
2
,:]
3
6
3
39. Determine a solution to each of the following linear systems. using the fact that Ax = h is consistent if and only
if h is a linear combination of the columns of A:
.
(, )
32. Write the following linear system in matrix fonn:
- 2xl
+ 3X2
XI -
5X2
= 5
= 4
(b)
40
Chapter 1
Linear Equations and Matrices
Key Terms
Properties of matrix :lddition
Zero matrix
Properties of matrix multiplication
-e-
Properties of scalar multiplication
Properties of transpose
Exercises
I I. Find two unequal 2 x 2 m:ltrices A and B such that
I. Prove Theorem 1.I(b).
2. Prove Theorem 1.I (d).
All -
3. Verify Theorem 1.2(a) for the following matrices:
12. Find two different 2 x 2 matrices A such that A" = O.
[~
A=
[~ ~l
B~
-I
[
3
1 - 3
13. Prove Theorem 1.3(a).
14. Prove Theorem 1.3(b"1.
-a
15. Verify Theorem 1.3(b) for r
and
[~
4. Prove Theorem 1.2(b) and (c)
16. Prove Theorem 1.3(c).
5. Verify Theorem I.I(c) lor the following matrices:
17. Ver ify Theorem l.3(c) lor r = - 3.
A=
[~
-3
- I
C =
6. Let A =
[aij ] be the I!
and(lij =Oifi i=then AB = kB.
"~[~
-;]
[ _~
1
3
-2
matrix defined by (Iii = k
j. Show that if B is any n x /I matrix.
/I
[CJ
C2
B~
and
.
L:>iAj.
[
- I
1
3
- 3
r = - 3.
20. The m:ltrix A contairt~ the weight (in pounds) of objects
packed on boMd a spacecraft on earth. The objects are
to be used on the moon where things weigh :lbout as
much. Write an expression kA that calculates the weight
of the objec ts on the moon.
!
whereA j is the jth row of A.
cosO
- si n O
1
18. Prove Theorem 1.3(d·..
j= l
LetA=[
[ ~ ;].
19. Verify Theorem 1.3(d) for the following m:ltrices:
x
CA =
.
B~
and
-!].
7. Let A be:lnm x II matrix :lnd C =
a I x II! matrix. Prove that
8
- 2. and A
4. s
sinO]
cosO .
21. (a ) A is a 360 x 2 matrix. The first column of A is
cos OQ. cos 10..... cos 359 :lnd the second column
is si n 0 °. sin 10 ..... si n 359 The graph of the ordered pairs in A is :I circle of radius I centered :It
the origin. Write an expression kA for ordered pairs
whose gr:lph is a circle of radius 3 centered at the
origin.
Q
:
Q
•
(a ) Determine a simple expression for A 2.
(b) Determine:l simple expression for A ].
(c) Conjecture the form of a si mple expression for
k a positive integer.
A".
(d ) Prove or dispro\le your conjecture in part (c).
9. Find a pair of unequal 2 x 2 matrices A and B . other than
those given in Example 9. suc h that AB = O.
10. Find two different 2 x 2 matrices A such that
A2 =
[~ ~l
(b ) Explain how to prove the claims about the circles in
p:lrt (:I).
22. Determine a scalar r such th:lt Ax = rx . where
1.5
25. (a) Show that ir A is an upper triangular matrix. then
AT is lower triangular.
(b) Show that if A is a lower triangular matrix. then AT
is upper lriangul:lr.
26. If A is a skew symmetric m.atrix. whal Iype of malrix is
AT? Justify your answer.
27. Show that if A is skew sym m~t ric, then the elements on
lhe main dia gonal of A are all 1.ero.
Special Types of Matrices and Pa rtitioned Matrices
Find the solutio n x.
38. The linear system A ~ .~ = b is such that A is nonsingular
wi lh
Find the solution x.
39. The linear system AT x = h is such that A is nonsingular
wit h
Al= [~
28. Show that if A is skew symllletric, the n A' is skew sy m·
metri c for any positive odd inlCger k.
29. Show 1hat if A is an It x II ma\Jix. then A = S + K . where
S is sy mmetric and K is skew sy mmetric. A lso show that
this decomposition is unique. (Hilll : Use Exercise 22.)
30. Let
: -n·
31. Show that the m:l1rix A =
32. IfD =
[~
[! !]
Find the solution
=
is singular.
o
(a)
.
o
(b) A
34. If A is a nonsingul ar matrix whose inverse is
x.
x.
. .'b [5]
( b ) Fmd a solutIOn 11
n
= [~
b =[ _~] .
Finda sOlutiOnif b =[~].
-2
[! ;]
and
41. Consider th e linear syMem A x = h. where A is the mao
trix defined in Exercise 33(a).
33. Find the inverse of each of the following matrices:
(a) A
~]
40. The linear system C T Ax = b is such that A and C are
nonsingular. wi th
Find the solution
Find the matrices Sand K desc ribed in Exercise 29.
53
=
6 .
42. Find t...."O 2 x 2 singula r matrices whose sum is nonsin·
gular.
[~
:l
43. Find twO 2 x 2 nonsUlgular matrices whose sum ii sin·
gular.
44. Pro\'e Corollary I. L
fi nd A.
45. Pro\'e Theorem 1.7.
35. If
and
B-
1
-- [ 3'
fi nd (AB )- I.
46. Prove Ihal if one row (column) o f the n X II matrix A con·
sists e nti rely of zeros. lhen A is singular. ( Hinl : Assume
lhal A is nonsingular; that is, th ere exists an /I x /I matrix
B such lhm AB = BA = I". E~labli s h aconlradiclion.)
47. Prove:
36. Suppose that
A-
I
=[:
~l
Solve the linear system Ax = h for each of the following
matrices b:
37. The linear sys te m AC x
nonsi ngul ar with
II is such that A and Care
If A is a diagona l illlitrix with nonzero di·
agonal el11ries {/11.{/ll ••••• II" • • then A is nonsingu·
lar and A- I is a dillgonal malrix Wilh diagonal en tries
1 / 11 1 1. l / lIll ..... 1/,,"".
48. Lo< A
= [~
o
-3
o
49. For an /I x /I diagonal matrix A whose diagonal entries
arc lIll' li n . .... a,,", compute AI' for a nonnegative inte·
ge r fJ.
50. Show Ihat if A B
1J =c.
AC and A is
nonsin~ular.
then
94
Chapter 2 Solving linear Systems
Remark The procedure given here fo r fi nding a matri x K in reduced row echcion limn that is row equi valent to a given matrix A is not the only one possible.
For example, instead of first obtaining a matrix H in row echelon form that is row
equivalent to A and then transfonning H 10 reduced row echelon form , we could
proceed as follows. First, zero out the entries below a leadi ng I and then immediately zero out the entries above the lead ing I. This procedure is not as efficient as
the procedure given in Example 6.
Key Terms
Elimination method
Reduced row echelon form
Leading one
w,.
Row echelon fonl1
Elementary row (column) operation
Row (column) equivalent
Exercises
l. Find a row echelon fonn of each of the given matrices.
Record the row operations you perform. using the notalion for elementary row operations.
(a) A =
(b) A =
n=; -n
[ ; ~ =:]
5
- 2
6
- 2
(a) A =
~3
Ih)
2
A~
I
- I
4
I
I
-6
-4
-2
[ - I
- 3
4
H
- I
0
3]
10
- 14
[~
A~[~
2
I
0
A~
3
-n
4
3
0
I
Ih)
0
0
0
0
4. Each of the given matrices is in row echelon fonn. De-
-1]
[~
A~[~
0
I
0
0
3
0
~]
-3
I
0
0
0
2
- I
~]
5. Find the reduced row echelon fonn of each of the given
matrices. Record the row operations you perfonn . using
Ihe notation for elementary row operations.
I' J
A
-4]
10
- 12
3. Each of the given matrices is in row echelon fonn. Delermine its reduced row echelon fonn. Record the row
operati ons you perform. using the notation for elemenmry row operations.
(a) A =
tennine its reduced row echelon fonn. Record the row
operations you perform. using the notation for elementary row operations.
(. )
2. Find a row echelon fonn of each of the gi ven matrices.
Record the row operations you perfonn. using the notalion fOt demelHary row operal;olls.
Ih)
Pivot column
Pivot
(h) A
~
H !-n
~ U~ ~~]
6. Find the reduced row echelon fonn of each of the given
matrices. Record the row operations you perfonn . uSing
the notation for elementary row operations.
(a) A =
[-i =; -n
(b ) A =
[ ; ~ =:]
5
6 - 3
- 2 - 2
2
7. Let x. y . z. and w be nonzero real numbers. Label each of
2.2 Solving Li near Systems
Ihe following matrices REF if it is in row echelon form.
RREF if it is in reduced row echelon form. or N if it is
[lot REF and not RREF'
x )'
I
x
)'
(a)
[~
I
(e)
~
[
1
o
o
w
1
o 1
o o
o o
x
1
)'
w
o
o
o
o o o 1
o o o o
o~]
(a) Every matrix is row equivalent to itself.
(b ) If B is row equivalent to A . then A is row equi"alent
to B .
(c)
If C is row equil'alent to Band B is row equil'alent
to A. then C is row equivalent 10 A .
II. Let
~]
[~
10. Prove:
1
8. Le t x . y . ; . and w be nonzero rt!al numbers. Labe l each of
the fo llowing ma trices REF if it is in row echelon form.
RREF if it is in reduced row echelon form. or N if it is
[lot REF and not RREF:
x 0
0
0 1
(b)
(a)
1
0 0
0
0 0
[~ ;]
[~ ~]
r]
2
- 3
o
1
3
2
3
0
-~].
-3
(a) Find a matrix in column echelon form that is column
equivalent to A.
(b) Find a ma trix in reduced column echelon form that
is col umn eq uivalent to A.
12. Repeat Exercise II for the matrix
1
lJ
0
13. Determine the reduced row echelon form of
2
)
(e.)
9. Let A be an /I x II matrix in reduced row echelon form.
Prove that if A of- In. then A has a row consisting entire ly
of zeros.
m
95
A~
3
4
3
- I
2
4
cosO SinO]
[ -sin O cos O .
Solving Lint!ar Systt!ms
In this secti o n we use the echelon form s developed in Section 2. 1 to more efficiently dete rmine the solution o f a linear system compared w ith the elimination
method o f Section 1.1 . Usi n g the augmented matrix of a linea r system together
with an ec helon form. we develop two me thods fo r solvin g a system o f III line ar
e quations in /I unk now ns. These me thods take the augmented mat rix of the line ar
system, perform e leme ntary row operations o n it, and obtain a new matrix th at
represents an eq ui valent linear system (i .e., a system that has the same solutions as
the origi nal linear system). T he important poi nt is that the latter linear system can
be solved more easily.
To sec how a linear system whose augme nted mat rix has a particular fo rm can
be read ily solve d . su ppose that
2
o
oi
3]
I: 2
I :- 1
re prese nts the augmented matrix o f a linear system. The n the solutio n is quic kly