matrices & systems of linear equations
DESCRIPTION
Matrices & Systems of Linear Equations. Special Matrices. Special Matrices. Equality of Matrices. Two matrices are said to be equal if they have the same size and their corresponding entries are equal. Equality of Matrices. Use the given equality to find x, y and z. - PowerPoint PPT PresentationTRANSCRIPT
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Matrices & Systems of Linear Equations
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Special Matrices
000
000
000
,0
0
0000,00
00
:
3321
1422
OO
OO
Examples
MatrixZero
0214
4513
8102
8101
453
912
601
,25
01,5
:Examples
MatrixnbynAn
MatrixSquare
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Special Matrices
8000
0500
0070
0001
400
010
001
,20
01,5
:Examples
zeroesarediogonal
maintheonnotarethatentriesall
ifmatrixsquareA
MatrixDiagonal
1000
0100
0010
0001
100
010
001
,10
01,1
:
44
332211
I
III
Examples
onesarediogonal
maintheonarethatentriesall
ifmatrixdiagonalisA
MatrixIdentity
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Equality of Matrices
Two matrices are said
to be equal if they have
the same size and their
corresponding entries
are equal
45
21,
43
21
?
584
573
062
951
,
5509
8765
4321
?
equalmatricesfollowingtheAre
equalmatricesfollowingtheAre
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Equality of Matrices
Use the given equality
to find x, y and z
52343
21
.2
509
8735
4221
5509
8765
4321
.1
25
z
yx
z
y
x
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Matrix Addition and SubtractionExample (1)
12915
1296
471
391887
664524
135201
318
642
150
987
654
321
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Matrix Addition and SubtractionExample (2)
671
012
231
391887
664524
135201
318
642
150
987
654
321
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Multiplication of a Matrix by a Scalar
21176
666
512
318
642
150
181614
12108
642
318
642
150
)1(
987
654
321
2
)2(
45535
30020
15105
)9(5)1(5)7(5
)6(5)0(5)4(5
)3(5)2(5)1(5
917
604
321
5
)1(
Example
Example
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Matrix Multiplication(n by m) Matrix X (m by k) Matrix
The number of columns of the matrix on the left
= number of rows of the matrix on the right
The result is a (n by k) Matrix
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Matrix Multiplication3x3 X 3x3
332211332211332211
332211332211332211
332211332211332211
333
222
111
321
321
321
zczczcycycycxcxcxc
zbzbzbybybybxbxbxb
zazazayayayaxaxaxa
zyx
zyx
zyx
ccc
bbb
aaa
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Matrix Multiplication1x3 X 3x3→ 1x3
32211332211332211
333
222
111
321
azazayayayaxaxaxa
zyx
zyx
zyx
aaa
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Example (1)
323
422
241
)2)(1()1)(1()0)(2()1)(1()1)(1()1)(2()1)(1()0)(1()2)(2(
)2)(2()1)(0()0)(0()1)(2()1)(0()1)(0()1)(2()0)(0()2)(0(
)2)(1()1(4)0(1)1)(1()1(4)1(1)1)(1()0(4)2(1
211
110
012
112
200
141
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Example (2)(1X3) X (3X3) → 1X3
52411
)4(4)10(3)3(2)3(4)2(3)1(2)2(4)1(3)0(2
432
1021
310
432
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Example (3)(3X1) X (1X2) → 3X2
00
76
3530
)7(0)6(0
)7(1)6(1
)7(5)6(5
76
0
1
5
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Example (4)
37164
1110
043
540
311
121
540
311
121
540
311
1212
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Transpose of Matrix
863
752
041
870
654
321
)1(
333
222
111
321
321
321
T
T
Example
cba
cba
cba
A
ccc
bbb
aaa
A
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30273
251214
9102
200
020
002
32273
251014
9104
100
010
001
2
24210
181512
963
863
752
041
2
870
654
321
3
870
654
321
)2(
3I
Example
T
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870
654
321
870
654
321
)3(
TT
Example
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Properties of the Transpose
TTT
TT
ABAB
AA
)(.2
)(.1
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Matrix ReductionDefinitions (1)
1. Zero Row: A row consisting entirely of zeros
2. Nonzero Row: A row having at least one nonzero entry
3. Leading Entry of a row: The first nonzero entry of a row.
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Matrix ReductionDefinitions (2)
Reduced Matrix: A matrix satisfying the following:
1. All zero rows, if any, are at the bottom of the matrix
2. The leading entry of a row is 1
3. All other entries in the column in which the leading entry is located are zeros.
4. A leading entry in a row is to the right of a leading entry in any row above it.
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Examples of Reduced Matrices
000
710
501
.3
000
010
001
.2
100
010
001
.1
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Examples matrices that are not reduced
.
2
000
001
010
.2
12
100
060
001
.1
itaboverowtheinentry
leadingtheofrightthetonotisrowinentryleadingThe
notisrowinentryleadingThe
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)3(
,12
000
210
031
.4
.
2
010
000
001
.3
columntheintheNotice
zerosarelocatedisitwhichin
columntheinentriesotherallnotbut
isrowinentryleadingThe
matrixtheofbottomtheatnotbut
rowzeroaisRow
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Elementary Row Operations
1. Interchanging two rows
2. Replacing a row by a nonzero multiple of itself
3. Replacing a row by the sum of that row and a nonzero multiple of another row.
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Interchanging Rows
225
320
1263
225
1263
32021 RR
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Replacing a row by a nonzero multiple of itself
225
320
421
225
320
12631
3
1R
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Replacing a row by the sum of that row and a nonzero multiple of another row
22120
320
421
225
320
421)5( 13 RR
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Augmented Matrix Representing a System of linear Equations
7
3
7
225
1263
320
:
7225
31263
732
:
matrixaugmentedthebydrepresenteIs
zyx
zyx
zy
equationslinearofsystemThe
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29
)(
3
4
10
01
9
4
20
01
9
5
20
21
6
5
43
21
:
6
5
43
21
:
Re
)1(
221
21
12
R
RR
RR
Solution
operations
rowbasicapplyingbymatrixaugmentedfollowing
theofrighttheonmatrixtwobytwotheduce
Example
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Solving a System of Linear Equations by Reducing its Augmented Matrix
Using Row Operations
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29
29
&4
:
4
10
01
:
6
5
43
21
:
:
:
643
52
:
)1(
yx
thatconcludeWe
matrixtheatarrivingmatrixthisreduceWe
operations
matrixaugmentedtheconstructWe
Solution
yx
yx
equationslinearofsystemfollowingtheSolve
Example
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7
3
7
225
1263
320
:
7225
31263
732
:
)2(
matrixaugmenteditsreducingby
zyx
zyx
zy
equationslinearofsystemfollowingtheSolve
Example
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Solution
2
8
22120
10
701
2
1
22120
10
421
2
7
1
22120
320
421
7
7
1
225
320
421
7
7
3
225
320
1263
7
3
7
225
1263
320
27
23
)2(
27
23
2
1
)5(3
1
212
131
21
RRR
RRR
RR
![Page 36: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/36.jpg)
1
2
1
100
010
001
1
1
100
10
001
1
8
100
10
701
40
8
4000
10
701
2
8
22120
10
701
)(
27
23
7
27
23
40
1
27
23
12
27
23
323
2
313
23
RR
RRR
RR
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Solution of the System
1
2
1
,
1
2
1
100
010
001
z
y
x
Thus
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The Idea behind the Reduction Method
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7
3
7
225
1263
320
:matrixaugmentedThe
7225
31263
732
:
zyx
zyx
zy
equationslinearofsystemThe
![Page 40: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/40.jpg)
Interchanging the First & the Second Row
7
7
3
225
320
1263
7
3
7
225
1263
32021 RR
7225
732
31263
7225
31263
732
:)2()1(
zyx
zy
zyx
zyx
zyx
zy
EqandEqBetweem
placesthengtIntercangi
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Multiplying the first Equation by 1/3
7
7
1
225
320
421
7
7
3
225
320
1263
13
1R
7225
732
142
7225
732
31263
zyx
zy
zyx
zyx
zy
zyx
![Page 42: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/42.jpg)
Subtracting from the Third Equation 5 times the First Equation
2
7
1
22120
320
421
7
7
1
225
320
421
)5( 13 RR
22212
732
142
7225
732
142
zy
zy
zyx
zyx
zy
zyx
![Page 43: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/43.jpg)
Subtracting from the First Equation 2 times the Second Equation
2
8
22120
10
701
2
1
22120
10
421
27
23
)2(
27
23
21 RR
222122
7
2
3
87
222122
7
2
3
142
zy
zy
zx
zy
zy
zyx
![Page 44: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/44.jpg)
Adding to the Third Equation 12 times the Second Equation
40
8
4000
10
701
2
8
22120
10
701
27
23
12
27
23
23 RR
40402
7
2
3
87
222122
7
2
3
87
z
zy
zx
zy
zy
zx
![Page 45: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/45.jpg)
Dividing the Third Equation by 40
1
8
100
10
701
40
8
4000
10
701
27
23
40
1
27
23
3R
12
7
2
3
87
40402
7
2
3
87
z
zy
zx
z
zy
zx
![Page 46: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/46.jpg)
Adding to the First Equation 7 times the third Equation
1
1
100
10
001
1
8
100
10
701
27
23
7
27
23
31 RR
12
7
2
3
1
12
7
2
3
87
z
zy
x
z
zy
zx
![Page 47: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/47.jpg)
Subtracting from the Second Equation 3/2 times the third Equation
1
2
1
12
7
2
3
1
z
y
x
z
zy
x
1
2
1
100
010
001
1
1
100
10
001
)(
27
23
323
2 RR
![Page 48: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/48.jpg)
Systems with infinitely many Solutions
:
23
:,
23
:,
?)(
sec
642
32
:
:)1(
tableoppositetheinshownAs
rx
getwernumberrealanybeylettingBy
yx
equationoneonlyhaveweThus
Whyfirsttheas
sametheisequationondthethatNotice
yx
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systemfollowingtheSolve
Example
y = rx=3-2r
03
-15
11
10-17
![Page 49: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/49.jpg)
yx
yx
Thus
matrixtheatarrivingmatrixthisreduceWe
matrixaugmentedtheconsidersLet
RR
23
00
32
:
0
3
00
21
:
6
3
42
21
:'
12 )2(
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Systems with infinitely many Solutions
:
3:,
302
:),1(
033
:),3()1(
,
)3(
),2()1(
04
052
02
:
:)2(
tableoppositetheinshownAs
ryandrxgetwernumberrealanybezlettingThus
zxzzx
getweEqinthatngSubstituti
zyzy
getweEqfromEqgSubtractin
equationstindependentwoonlyhaveweThus
Eq
getweEqfromEqgsubtractinwhenthatNotice
zyx
zyx
zyx
systemfollowingtheSolve
Example
z=rx=-3ry=-r
000
1-3-1
-103010
1/3-1-1/3
![Page 51: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/51.jpg)
ryandrx
getwernumberrealanybezlettingBy
zyandzx
zyandzx
usgivesrowsotherThe
ormationanycontributenotdoesrowlastThe
matrixtheatarrivingmatrixthisreducesLet
matrixaugmentedtheconsidersLet
3
:,
3
003
:
00:inf
0
0
0
000
110
301
:'
0
0
0
411
512
121
:'
![Page 52: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/52.jpg)
Details of reduction
0
0
0
000
110
301
0
0
0
000
411
301
0
0
0
000
411
903
0
0
0
000
411
121
0
0
0
411
411
121
0
0
0
411
512
121
)(
3
1
)2()(
)(
12
1
2123
12
RR
R
RRRR
RR
![Page 53: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/53.jpg)
Systems with no Solution
solutionnohassystemtheThus
equationsbothsatisfyingyandxnumbersnoarethereThus
statementimpossiblethegetwefromEqEqgsubtractinBy
yxthatclaimsEqfirsttheBut
yx
bygMultiplyinnumberthebyEqontheDividing
WhyfirstthescontradictequationondthethatNotice
yx
yx
systemfollowingtheSolve
Example
10
:),3()1(
32:
)3(42
:)2
1(2sec
?)(sec
842
32
:
:)1(
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impossibleisWhich
yx
Thus
matrixtheatarrivingmatrixthisreduceWe
matrixaugmentedtheconsidersLet
RR
20
32
:
2
3
00
21
:
8
3
42
21
:'
12 )2(
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slutionnohassystemthe
systemtheofequations
threetheallsaisfyingzandyxnumbersrealnoarethereThus
statementimpossiblethegetweEqfromEqgSubtractin
zythatstateswhichEqscontradictequationThis
zy
getweEqfromEqgsubtractinwhenthatNotice
zyx
zy
zyx
systemfollowingtheSolve
Example
.
,,
20
:),4()2(
32:),2(
)4(52
),1()3(
12
32
642
:
:)2(
![Page 56: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/56.jpg)
10
02
0
)(
1
0
0
000
210
001
:'
1
3
6
211
210
421
:'
zy
x
solutionnohaswhichsystemthetoscorrespondThis
matrixtheatarrivingmatrixthisreducesLet
matrixaugmentedtheconsidersLet
![Page 57: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/57.jpg)
Details of the reduction
1
0
0
000
210
001
1
3
0
000
210
001
2
3
0
000
210
001
5
3
0
210
210
001
5
3
6
210
210
421
1
3
6
211
210
421
)3(
)2
1(
)()2(
)(
32
3
2321
13
RR
R
RRRR
RR
![Page 58: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/58.jpg)
solutionnohaswhich
zyx
zy
systemthetodscorrresponWhich
approachbetterA
RR
RR
12
32
20
:
1
3
2
211
210
000
1
3
5
211
210
210
1
3
6
211
210
421
:
)(
)(
21
31
![Page 59: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/59.jpg)
Finding the Inverse of an nXn square Matrix A
1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ]
2. Reduce [A| In ] to [In | B ] if possible
Then
B = A-1
![Page 60: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/60.jpg)
Example (1)
1
115
014
001
100
920
201
101
014
001
820
920
201
100
014
001
1021
920
201
100
010
001
1021
124
201
:
1021
124
201
32
31
21
)(
)(
)4(
1
RR
RR
RR
Solution
AFindALet
![Page 61: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/61.jpg)
115
4
229
,
115
4
229
100
010
001
115
9841
229
100
020
001
115
014
229
100
920
001
29
2411
29
241
)9(
)2(
221
23
13
A
Thus
R
RR
RR
![Page 62: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/62.jpg)
Example (2)
inversenohasAThus
Solution
AFindALet
RR
RR
RR
111
012
001
000
430
321
103
012
001
430
430
321
100
012
001
533
430
321
100
010
001
533
212
321
:
533
212
321
32
31
21
)(
)3(
)2(
1
![Page 63: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/63.jpg)
inversenohasAThus
WayAnother
RR
RR
111
011
001
000
533
321
100
011
001
533
533
321
100
010
001
533
212
321
:
32
21
)(
)(
![Page 64: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/64.jpg)
Inverse MatrixThe formula for the inverse of a 2X2 Matrix
AAIAA
thatCheck
ac
bd
AA
ThenAIf
bcadA
dc
baA
Let
12
1
1
:
det
1
:,0det
det
&
93
52
3
21
3
53
10
01
3
21
3
53
93
52
3
21
3
53
23
59
3
1
23
59
det
1
03151893
52det
93
52:
:
1
Checking
AA
A
ALeT
Example
![Page 65: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/65.jpg)
Using the Inverse Matrixto Solve System of Linear Equations
12
:
,
1
2
)15)(3
2()9)(1(
)15)(3
5()9(3
15
9
3
21
3
53
93
52
3
21
3
53
15
9
93
52
15
9
93
52
:Re
1593
952
:
2
yandx
isSolutionThe
Thus
y
x
y
xI
y
x
y
x
yx
yx
formmatrixinwriting
yx
yx
equationslinearofsystemfollowingtheSolve
![Page 66: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/66.jpg)
Problem
101
124
113
212
123
112
:
122
223
12
:
1
thatgivenIf
zyx
zyx
zyx
equationslinearofsystemfollowingtheSolve
![Page 67: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/67.jpg)
21,2
:
,
2
1
2
2
1
2
1
2
1
101
124
113
212
123
112
101
124
113
1
2
1
212
123
112
1
2
1
22
23
12
:Re
122
223
12
3
zandyx
isSolutionThe
Thus
z
y
x
z
y
x
I
z
y
x
z
y
x
zyx
zyx
zyx
formmatrixinwriting
zyx
zyx
zyx
![Page 68: Matrices & Systems of Linear Equations](https://reader030.vdocuments.mx/reader030/viewer/2022033015/56813551550346895d9cb0e2/html5/thumbnails/68.jpg)
Homework
6.1 Examples: 3 Exercises: 17 - 20
6.2 Examples: 1, 2, 4, 5 and 6 Exercises: odd numbered:1—17 and 25,29,35,37,39,41
6.3 Examples: 1, 2, 3, 4, 5, 7, 10, 11, 12 and 13. Exercises:19, 21, 23, 25, 27, 31, 33, 37, 51, 53, 57, 59, 61
6.5 Examples: 2, 3.a and 4. See given exercises.
6.6 Examples: 1 - 6 Exercises: odd numbered:1—15, 21, 27, 29, 35 and 37.
1.