| Date post: | 02-Jun-2018 |
| Category: |
Documents |
| Upload: | fuckthisscribd |
| View: | 214 times |
| Download: | 0 times |
of 77
8/10/2019 Linear and Its Applications - Lay: Chapter 10
1/77
Addison-Wesley
8/10/2019 Linear and Its Applications - Lay: Chapter 10
2/77
0
8/10/2019 Linear and Its Applications - Lay: Chapter 10
3/77
8/10/2019 Linear and Its Applications - Lay: Chapter 10
4/77
8/10/2019 Linear and Its Applications - Lay: Chapter 10
5/77
WEB
8/10/2019 Linear and Its Applications - Lay: Chapter 10
6/77
j i
P
j i .i;j / P m
12 : : : m m m
P D
j m26664
:::
#
pij !
37775
i
m
n
n m
n D
26666664
a1:::
aj:::
am
37777775
j n
8/10/2019 Linear and Its Applications - Lay: Chapter 10
7/77
1
0
j P
j
nC1 D Pn
n D 1 ; 2 ; : : :
n D Pn0
n 0
P
P
nC1 P n n n C 1
P
p
1 p
01
P D
p
1 p
1 p
p
p D :99
8/10/2019 Linear and Its Applications - Lay: Chapter 10
8/77
1 p
p p
1 p
0 1
0 D
1
0
2 D P20 D
:99 :01
:01 :99
21
0
D
:9802 :0198
:0198 :9802
1
0
D
:9802
:0198
:9802
.:99/2 D :9801
k
2k
n C 1
n
k D 3 6
01 2 3 4 5 6
0 n 1
n C 1 6 n
5 n C 1 P P 06
0 D
2666666664
0
1
0
0
0
0
0
3777777775
6 D
2666666664
0
0
0
0
0
1
0
3777777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
9/77
i n 0 < i < 6
i 1i C 1 n C 1 ii 1
i
i =6 i i C 1
6 i .6 i/=6 i 15 P
k D 3
P D
2666666664
0
1
0
0
0
0
0
1=6
0
5=6
0
0
0
0
0
1=3
0
2=3
0
0
0
0
0
1=2
0
1=2
0
0
0
0
0
2=3
0
1=3
0
0
0
0
0
5=6
0
1=6
0
0
0
0
0
1
0
3777777775
Z a q s X 1
1
0 1 2 3 4 5 6
f: : : g
f1 ; 2 ; : : : ; ng
p
1 p
p D 1=2 p 1=2
... ...
p p p p
1 p 1 p 1 p 1 p
k 2 k 1 k k+ 1 k+ 2
2; : : : ; n 1
1 n
1n
1 n
8/10/2019 Linear and Its Applications - Lay: Chapter 10
10/77
f1; 2; 3; 4; 5g
P D
26666664
1
0
0
0
0
p
0
1 p
0
0
0
p
0
1 p
0
0
0
p
0
1 p
0
0
0
0
1
37777775
1 1 1
5 1 5 f1; 2; 3; 4; 5g
P D
26666664
0
1
00
0
p
0
1 p0
0
0
p
01 p
0
0
0
p0
1 p
0
0
01
0
37777775
1 1 2
5 1 4
n
n D 7
0 7
f0;1;2;3;4;5;6;7g
p D 1=2
1
7
5
1=2 1=2 6
1
3 42
5 6
7
8/10/2019 Linear and Its Applications - Lay: Chapter 10
11/77
P D
2666666666664
0
1=2
1=2
0
0
0
0
1=3
0
1=3
0
1=3
0
0
1=4
1=4
0
1=4
0
1=4
0
0
0
1
0
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
0
1
0
3777777777775
6 2
3 D P30 D P
3
2666666664
0
00
0
0
1
0
3777777775D
2666666664
:0833
:0417:4028
0
:2778
0
:1944
3777777775 6 2 :0417
3
1 2
3
5
4
1 2
3
5
4
P D
26666664
0
1=2
1=2
0
0
1=3
0
1=3
1=3
0
1=4
1=4
0
1=4
1=4
0
1=3
1=3
0
1=3
0
0
1=2
1=2
0
37777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
12/77
5 D P50 D P
5
2
666640
0
1
00
3
77775 D2
66664:1507
:2143
:2701
:2143:1507
3
77775 3 :2701
1
342
5 6
7
6
1=3 3 5 7
j i j
i
5
3
P D
2666666666664
0
0
1
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
1
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
0
0
1
3777777777775
4 7
p44 p77 1
1 n
n C 1
8/10/2019 Linear and Its Applications - Lay: Chapter 10
13/77
4 7 4
4 D P40 D
2
666666664
:1319
:0833
:0880
:1389:2199
:0833
:2546
3
777777775 3 :0880
4 7
n n
fng n
P D
24 :1 :2 :3 :3
:6 :2
35
p D :97
P
P
P D
:3 :4
:7 :6
P D
:3 :7
:4 :6
P D
1 :5
0 :5
P D
:2 1:1
:8 :1
3 12 P
3
P D
:6 :5
:4 :5
0 D
1
0
P D
:3 :8
:7 :2
0 D
:5
:5
P
01
0 n D 0
1 n
P D
1=3 3=4
2=3 1=4
n D 3
P D
:4 :2
:6 :8
n D 5
P
0 1 2
0 n D 0
1 n
P D24 1=3 1=4 1=21=3 1=2 1=4
1=3 1=4 1=4
35n D 2
P D
24 :1 :2 :4:6 :3 :4
:3 :5 :2
35n D 3
8/10/2019 Linear and Its Applications - Lay: Chapter 10
14/77
f1; 2; 3; 4g
2 3
f1; 2; 3; 4g
p D :2
2 3
1 2
4 3
5
1 2
34
1 2
3 4
1 4
2 5
3
3
4 5
1 2
1 2 3
54
1 32
4 65
3
4 5
1 2
P
. i; j / P
j i
fng nC1
n
8/10/2019 Linear and Its Applications - Lay: Chapter 10
15/77
. i; j / P3
i j
2
2
:58 0
2
p
p D :99
k 2k
p
1 p
p D 1 k D 3
k D 3 p D 1=2
3 5
2k
k
k
k
j
0 < j < k
j 1 .j=k/2
j C 1
..k j/=k/2
k D 5
k D 5
8/10/2019 Linear and Its Applications - Lay: Chapter 10
16/77
p
p D :6
p
q
p D q D :6
p
q
p D q D :6
P
p Pn
pn D 1 ; 2 ; : : :
1
P D
24 :1 :5 :2:3 :3 :3
:6 :2 :5
35
P D
:97 :03
:03 :97
0 D
0
1
2 D P30 D
:9153 :0847
:0847 :9153
0
1
D
:0847
:9153
:0847
8/10/2019 Linear and Its Applications - Lay: Chapter 10
17/77
n n
fng
fng
n Pn
n
P D
24 :5 :2 :3:3 :8 :3
:2 0 :4
35
0 D
24 10
0
35 n D
24 :3:6
:1
35
n!1
n D
P
P2 D
2
4:3700 :2600 :3300
:4500 :7000 :4500
:1800 :0400 :2200
3
5 P3 D
2
4:3290 :2820 :3210
:5250 :6500 :5250
:1460 :0680 :1540
3
5P4 D
24 :3133 :2914 :3117:5625 :6250 :5625
:1242 :0836 :1258
35 P5 D
24 :3064 :2958 :3061:5813 :6125 :5813
:1123 :0917 :1127
35
P10 D
24 :3002 :2999 :3002:5994 :6004 :5994
:1004 :0997 :1004
35 P15 D
24 :3000 :3000 :3000:6000 :6000 :6000
:1000 :1000 :1000
35
fPng n
P D
P P
P D
P
1
P
1
8/10/2019 Linear and Its Applications - Lay: Chapter 10
18/77
n!1
n n
n
n!1n D
n!1n D
n!1
n D
n n
n
q1 D :3
q2 D :6
q3 D :1 q
N n n N
:3
:6 :1
f1; 2; 3; 4; 5g
P D
2666640
1=21=2
0
0
1=3
01=3
1=3
0
1=4
1=40
1=4
1=4
0
1=31=3
0
1=3
0
01=2
1=2
0
377775
P D
.P I / D
266664
1 1=3 1=4 0 0 0
1=2 1 1=4 1=3 0 0
1=2 1=3 1 1=3 1=2 0
0 1=3 1=4 1 1=2 0
0 0 1=4 1=3 1 0
377775
266664
1 0 0 0 1 0
0 1 0 0 3=2 0
0 0 1 0 2 0
0 0 0 1 3=2 0
0 0 0 0 0 0
377775
q5
266664
1
3=2
2
3=2
1
377775
q5
8/10/2019 Linear and Its Applications - Lay: Chapter 10
19/77
D 1
7
266664
1
3=2
2
3=2
1
377775
D
266664
1=7
3=14
2=7
3=14
1=7
377775
266664
:142857
:214286
:285714
:214286
:142857
377775
q
P
P10 D
266664
:144169 :141561 :142613 :144153 :142034
:212342 :216649 :214286 :211922 :216230
:285226 :285714 :286203 :285714 :285226
:216230 :211922 :214286 :216649 :212342
:142034 :144153 :142613 :141561 :144169
377775
P10
f1; 2; 3; 4; 5g
P D
266664
1
0
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
0
1
377775
23 4
n Pn
P20 D266664
1 :74951 :49951 :24951 0
0 :00049 0 :00049 0
0 0 :00098 0 0
0 :00049 0 :00049 0
0 :24951 :49951 :74951 1
377775
P30 D
266664
1 :749985 :499985 :249985 0
0 :000015 0 :000015 0
0 0 :000030 0 0
0 :000015 0 :000015 0
0 :249985 :499985 :749985 1
377775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
20/77
Pn
266664
1 :75 :5 :25 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 :25 :5 :75 1
377775
n
1 5
0 q 1 266664
q
0
0
0
1 q
377775
P
n
0
f1; 2; 3; 4; 5g
P D
266664
0
1
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1
0
377775
n 1 1 n
2 n
n n
2
n n
Pn Pn
n
P20 D
2
66664:2505 0 :2500 0 :2495
0 :5005 0 :4995 0
:5000 0 :5000 0 :5000
0 :4995 0 :5005 0:2495 0 :2500 0 :2505
3
77775
P21 D
266664
0 :2502 0 :2498 0
:5005 0 :5000 0 :4995
0 :5000 0 :5000 0
:4995 0 :5000 0 :5005
0 :2498 0 :2502 0
377775
Pn P
8/10/2019 Linear and Its Applications - Lay: Chapter 10
21/77
2
66664
1=8
1=4
1=4
1=4
1=8
3
77775 P
f1; 2; 3; 4; 5g
P D
266664
1=4
1=4
1=2
0
0
1=3
1=3
1=3
0
0
1=2
1=4
1=4
0
0
0
0
0
1=3
2=3
0
0
0
3=4
1=4
377775
1 2 3
45
Pn 266664
4=11
3=11
4=11
0
0
377775
266664
0
0
0
9=17
8=17
377775
12 3 45
n P
n!1
Pn D
0
n!1
Pn0 D n!1
n D
P
1
Pn
n
P
8/10/2019 Linear and Its Applications - Lay: Chapter 10
22/77
Pk
P
P Pk
Pk k
k k
m n m n
. i; j / .i;j /
P m m m 2
n!1
Pn D
0 n!1
Pn0 D
P
1
P 1 jj < 1
P Pn
D
24 :3:6
:1
35
P P
8/10/2019 Linear and Its Applications - Lay: Chapter 10
23/77
1
342
5 6
7
P
P D
2
666666666664
0
0
1
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
1
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
0
0
1
3
777777777775
4 7
n
n ! 1
P
P j j P 26664
1=n
1=n:::
1=n
37775
n P
P D
2
666666666664
0
0
1
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
1=7
1=7
1=7
1=7
1=7
1=7
1=7
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
1=7
1=7
1=7
1=7
1=7
1=7
1=7
3
777777777775
j i i j
8/10/2019 Linear and Its Applications - Lay: Chapter 10
24/77
i
j Pk
P
p 01 j p
j 1 p
P
G D pP C .1 p/K
K n n 2
66641=n
1=n:::
1=n
3
7775 G G
G1 D G p 0 1
p D :85
G D :85
2666666664
0 1=2 0 1=7 0 0 1=7
0 0 1=3 1=7 1=2 0 1=7
1 0 0 1=7 0 1=3 1=7
0 0 1=3 1=7 0 0 1=7
0 1=2 0 1=7 0 1=3 1=7
0 0 1=3 1=7 1=2 0 1=7
0 0 0 1=7 0 1=3 1=7
3777777775
C :15
2666666664
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
1=7 1=7 1=7 1=7 1=7 1=7 1=7
3777777775
D
2666666664
:021429 :446429 :021429 :142857 :021429 :021429 :142857
:021429 :021429 :304762 :142857 :446429 :021429 :142857
:871429 :021429 :021429 :142857 :021429 :304762 :142857
:021429 :021429 :304762 :142857 :021429 :021429 :142857
:021429 :446429 :021429 :142857 :021429 :304762 :142857:021429 :021429 :304762 :142857 :446429 :021429 :142857
:021429 :021429 :021429 :142857 :021429 :304762 :142857
3777777775
3 K
8/10/2019 Linear and Its Applications - Lay: Chapter 10
25/77
D
2
666666664
:116293
:168567
:191263
:098844:164054
:168567
:092413
3
777777775 3
326514 7
G 1
f1; 2; 3g
P D24 1=2 0 1=21=2 1=2 0
0 1=2 1=2
35 P
f1; 2g
P
P
P D
:2 :4
:8 :6
P D
1=4 2=3
3=4 1=3
f1; 2; 3g
P
P
P D
24 1=3 1=4 01=3 1=2 1
1=3 1=4 0
35 P D
24 :1 :2 :3:2 :3 :4
:7 :5 :3
35
Pn n
P D
1=4 2=3
3=4 1=3
P D
24 1=4 3=5 01=4 0 1=3
1=2 2=5 2=3
35
8/10/2019 Linear and Its Applications - Lay: Chapter 10
26/77
P D
24 1=3 0 1=21=3 1=2 1=2
1=3 1=2 0
35
P D2664
1=2 0 1=3 0
0 2=5 0 3=7
1=2 0 2=3 0
0 3=5 0 4=7
3775 4
5
f1; 2; 3; 4g
f1; 2; 3; 4g p D :2
1 2
4 3
5
1 2
34
1 2
3 4
1 4
2 5
3
3
4 5
1 2
1 2 3
54
1 32
4 65
D
2
6666664
0
0
0
00
1
3
7777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
27/77
3
4 5
1 2
D 1 P P
P Pn
n
n!1
n D
1 2 5
3 4
1 4
2 5 6
3
1=2
1=2
1=2
1=2
1=4 1=4
1=2
2k
p
k D 3 p D 1=2
k D 3 p D 1=2
p0 < p < 1
k D 5
k D 5
0 q 1
266664
q
0
0
0
1 q
377775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
28/77
266664
1=8
1=4
1=4
1=4
1=8
377775
P20 P21
266664
4=11
3=11
4=11
0
0
377775
266664
0
0
0
9=17
8=17
377775
0 pq 1
P D p 1 q
1 p q
1p C q 1 P
pqP
P
0 pq 1
P D
24 p q 1 p qq 1 p q p
1 p q p q
35
p q P
P P
A m m Rm
D A
jy1j C : : : C jymj jx1j C : : : C jxmj
P
D 1 P
P D 1
P
1 jj < 1
1
P n n 1 D 1 : : :
n j1j > j2j j3j : : : jnj
0 D c1 C c22 C C cnn
P
k D Pk0
k c1 j2j fk g c1
P2 D
24 1=4 1=4 1=21=2 1=4 1=4
1=4 1=2 1=4
35
P k D 2
P D .P I / D
P I D
241=2 0 1=21=2 1=2 0
0 1=2 1=2
35
241=2 0 1=2 01=2 1=2 0 0
0 1=2 1=2 0
35
24 1 0 1 00 1 1 0
0 0 0 0
35
8/10/2019 Linear and Its Applications - Lay: Chapter 10
29/77
q3
24 11
1
35 q3 D
1=.1 C 1 C 1/ D 1=3
D 1
3
24 111
35 D 24 1=31=31=3
35 1=3
1=3
n
n!1
n D
n!1
n D
j i j
i i
j j i P
Pk k
Pk D
j m26664
:::#
pij !
37775
i
m
P
i j P i
j mn .j;i/ Pm . i; j / Pn
i j i j
m j i n
m n .i;i/ P0 D I 1
8/10/2019 Linear and Its Applications - Lay: Chapter 10
30/77
.i;j /
.j;i/ i
j j i
i j j k
i k
i j j
i
i j j
k i k
f1; 2; 3; 4; 5g
P
P D
26666664
1
0
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
0
1
37777775
q q q
q q q
11 2 3 14 5
23 4
.2;3/ .3; 2/ .3;4/
.4;3/ P 23 3
4 2 4
1 5 1 1
1 5
1 1
2 2 3 4
3 2 3 4
4 2 3 4
5 5
8/10/2019 Linear and Its Applications - Lay: Chapter 10
31/77
15 23 4
1 5
PP2
P3
f1; 2; 3; 4; 5g f1g
f2; 3; 4g f5g
f1g f2; 3; 4g f5g
f1; 2; 3; 4; 5g
P P2 P3 P4
P D
26666664
0
1
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1
0
37777775
P2 D
26666664
1=2
0
1=2
0
0
0
3=4
0
1=4
0
1=4
0
1=2
0
1=4
0
1=4
0
3=4
0
0
0
1=2
0
1=2
37777775
P3 D
26666664
0
3=4
0
1=4
0
3=8
0
1=2
0
1=8
0
1=2
0
1=2
0
1=8
0
1=2
0
3=8
0
1=4
0
3=4
0
37777775
P4 D
26666664
3=8
0
1=2
0
1=8
0
5=8
0
3=8
0
1=4
0
1=2
0
1=4
0
3=8
0
5=8
0
1=8
0
1=2
0
3=8
37777775
q q q 1
1
1 2 3 4 5
.i;j /
i j
f1; 2; 3; 4; 5g
8/10/2019 Linear and Its Applications - Lay: Chapter 10
32/77
P D
26666664
1=4
1=4
1=2
0
0
1=3
1=3
1=3
0
0
1=2
1=4
1=4
0
0
0
0
0
1=3
2=3
0
0
0
3=4
1=4
37777775
a ~
~ ~~ a
23
!a
s
4 51
~
a
q
12 3 45
1 2
3 4 5
f1; 2; 3g f4; 5g
P
k Pk i
j .i;j/ .j;i/ Pk
i j j i k i j
i j
8/10/2019 Linear and Its Applications - Lay: Chapter 10
33/77
.8 .6
1 51
.2
1
.6
3
2
.3 .1
4
.4
12 45
1 2 3 4 5
2 30 45
12 3 4 30
3
3
f1; 2g f3g f4; 5g
qi
i f1; 2; 3g
D
24 :2:5
:3
35 2
2 1=:5
2 1=5 1 1
i j
nij i
j nij
n ij nij
8/10/2019 Linear and Its Applications - Lay: Chapter 10
34/77
X x1; x2; : : :
EX D x1P .X D x1/ C x2P .X D x2/ C D
1
XkD1
xk P .XD xk /
P .X D xk / X
xk
ti i D Eni i ni i
i
i
i ti i
1=ti i
i qi
i qi
nij
i
j ti i D Eni i
ti i D 1
qi
qi i
t11D 1=:2 D 5 t22D 1=:5 D 2 t33D
1=:3 D 10=3 t33
tij D Enij i j
f1;2;3;4g
P D
2664
1=4 1=3 1=2 0
0 1=3 0 1=3
3=4 0 1=2 1=3
0 1=3 0 1=3
3775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
35/77
f1 ; 2 ; : : : ; ng
24 1=4 0 1=31=2 1 1=3
1=4 0 1=3
35
24 1=4 1=2 1=31=2 1=2 1=3
1=4 0 1=3
35
24 1 1=2 1=20 1=2 0
0 0 1=2
35
266664
0 0 0 1 1
1=3 0 0 0 0
2=3 0 0 0 0
0 1=4 2=3 0 0
0 3=4 1=3 0 0
377775
266666640 0 :4 0 :8 00 0 0 :7 0 :5
:3 0 0 0 :2 0
0 :1 0 0 0 :5
:7 0 :6 0 0 0
0 :9 0 :3 0 0
37777775
2666666664
0 1=3 0 2=3 1=2 0 0
1=2 0 1=2 0 0 1=3 0
0 2=3 0 1=3 0 0 2=5
1=2 0 1=2 0 0 0 0
0 0 0 0 0 0 3=5
0 0 0 0 1=2 0 0
0 0 0 0 0 2=3 0
3777777775
1 32
4 65
3
4 5
1 2
1 2 5
3 4
1 4
2 5 6
3
f1; 2; 3; 4g
f1; 2; 3; 4g
1 2
4 3
5
1 2
34
1 2
3 4
1 4
2 5
3
8/10/2019 Linear and Its Applications - Lay: Chapter 10
36/77
3
4 5
1 2
1 2 3
54
3
4 5
1 2
i j n
n 0 ij
.i ; j/ .j;i/ Pk
k ij
i j j
k i
k
1 2 5
3 4
1 4
2 5 6
3
k D 3
p D 1=2
k D 5
8/10/2019 Linear and Its Applications - Lay: Chapter 10
37/77
f1; 2; 3; 4; 5g
P D
266664
0 0 0 1=2 1
1=3 0 0 0 0
2=3 0 0 0 0
0 2=5 1=5 1=2 0
0 3=5 4=5 0 0
377775
1
1
.i;k/ PnCm
. i; j / Pm .j; k/
Pn
1 3 2 4
1 3 2
4 f1; 3g f2; 4g
C j
C i C k > 0 . i; j / Pk
C C
C C
D C
D i C D
i C D
f1; 2; 3; 4; 5g
f1; 2g f3g f4; 5g
f3g 3
8/10/2019 Linear and Its Applications - Lay: Chapter 10
38/77
.8 .6
1 51
.2
1
.6
3
2
.3 .1
4
.4
2 k D 1 f3g
f1; 2g 1
2 34 5
1 2 1 2 f1; 2g
f4; 5g
f1; 2; 3; 4; 5g
C1 C2
C1 C2
C1 C2 C1
C2 C1 C2
P D
26666664
1=4
1=4
1=2
0
0
1=3
1=3
1=3
0
0
1=2
1=4
1=4
0
0
0
0
0
1=3
2=3
0
0
0
3=4
1=4
37777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
39/77
f1; 2; 3g f4; 5g P
P D
P1 O
O P2
P1 D
2641=41=41=2
1=3
1=3
1=3
1=2
1=4
1=4
375
P2 D
"1=3
2=3
3=4
1=4
#
O
Pk D
"Pk1 O
O Pk2
#
k > 0 j i .i;j /
.j;i / Pk k > 0
P D
26666664
1=4
1=4
1=2
0
0
1=3
1=3
1=3
0
0
1=2
1=4
1=4
0
0
0
0
0
1=3
2=3
0
0
1=4
1=2
1=4
37777775
a ~
~ ~~ a
2
3
qa
s
4 51
~
a
q
q ~
f1; 2; 3g f4; 5g .3;5/
f4; 5g
f1; 2; 3g
P D
P1 S
O Q
P1
Q D
"1=3
2=3
1=2
1=4
#
S D
264
0
0
0
0
0
1=4
375
8/10/2019 Linear and Its Applications - Lay: Chapter 10
40/77
P1
f1; 2; 3g S
f4; 5g f1; 2; 3g Q
f4; 5g
Pk D
"Pk1 Sk
O Qk
#
Sk O Pk
f1; 2; 3g f1; 2; 3g
P
P D
266664
:8
:2
0
0
0
1
0
0
0
0
0
:3
:6
:1
0
0
0
0
0
1
0
0
0
:4
:6
377775
f1; 2g f3g f4; 5g
f1; 2g f4; 5g f3g
1 2 4 5 3
266664
:8
:2
0
0
0
1
0
0
0
0
0
:3
:6
:1
0
0
0
0
0
1
0
0
0
:4
:6
377775
!
266664
:8
:2
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
:4
:6
0
:3
:6
:1
0
377775
266664
:8
:2
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
:4
:6
0
:3
:6
:1
0
377775
!
266664
:8
:2
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
:4
:6
0
0
:3
:1
0
:6
377775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
41/77
P D
266664:8
:2
00
0
1
0
00
0
0
0
01
0
0
0
:4:6
0
0
:3
:10
:6
377775
DP
1 S
O Q
P
r P
P D
26664
P1 O:::
: : : ::: S
O PrO Q
37775
Pi i O
Q S P
Pk D
26664
Pk1 O:::
: : : ::: Sk
O PkrO Qk
37775
Sk Q S Sk
f1; 2; 3g
P D
2401
0
0
0
1
1
0
0
35
1 2
11
3
1
f1;2;3;4g
P D
266664
0
1
0
0
0
0
1=2
1=2
1
0
0
0
0
0
1
0
377775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
42/77
12 3 1 2
q q
1
1
3 4
1
3a C 4b a b
5
12 3 5 4
17
f1; 2; 3; 4; 5g
q q q 1
1 q q q
1 2 3 4 5
d d D 3 d D 2 d
d i
n i
i n
C C
d D 1
d D 1
d D 3
d D 2
8/10/2019 Linear and Its Applications - Lay: Chapter 10
43/77
P
P
P n n P k Pk
1 i; j n
a i j . i; j / Pa
b
j .j;j / Pm m b
PaCm D PaPm .i;j / PaCm
.i;j/ Pa .j;j/ Pm .i;j/
PaCm m b k
.i;j / a C b
. i; j / Pk .i;j /
Pk P
P P
P
n!1
Pn0 D
0
d > 1
P d > 1
0
n!1
1
d
PnC1 C C PnCd
0 D
d > 1
PnC10 PnC2
0 : : :
PnCd0 d > 1
d D 2
d > 1 n P
Pn !
26666664
1=4
0
1=2
0
1=4
0
1=2
0
1=2
0
1=4
0
1=2
0
1=4
0
1=2
0
1=2
0
1=4
0
1=2
0
1=4
37777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
44/77
PnC1 !
26666664
0
1=2
0
1=2
0
1=4
0
1=2
0
1=4
0
1=2
0
1=2
0
1=4
0
1=2
0
1=4
0
1=2
0
1=2
0
37777775
0
n!1
1
2
Pn C PnC1
0 D
266664
1=8 1=8 1=8 1=8 1=8
1=4 1=4 1=4 1=4 1=4
1=4 1=4 1=4 1=4 1=4
1=4 1=4 1=4 1=4 1=4
1=8 1=8 1=8 1=8 1=8
3777750 D
266664
1=8
1=4
1=4
1=4
1=8
377775
f1;2;3;4g
P D
2664
1=4 1=3 1=2 0
0 1=3 0 1=3
3=4 0 1=2 1=3
0 1=3 0 1=3
3775
f1 ; 2 ; : : : ; ng
24 1=4 0 1=31=2 1 1=3
1=4 0 1=3
35
24 1=4 1=2 1=31=2 1=2 1=3
1=4 0 1=3
35
24 1 1=2 1=20 1=2 0
0 0 1=2
35
266664
0 0 0 1 1
1=3 0 0 0 0
2=3 0 0 0 0
0 1=4 2=3 0 0
0 3=4 1=3 0 0
377775
26666664
0 0 :4 0 :8 0
0 0 0 :7 0 :5
:3 0 0 0 :2 0
0 :1 0 0 0 :5
:7 0 :6 0 0 0
0 :9 0 :3 0 0
37777775
2666666664
0 1=3 0 2=3 1=2 0 0
1=2 0 1=2 0 0 1=3 0
0 2=3 0 1=3 0 0 2=51=2 0 1=2 0 0 0 0
0 0 0 0 0 0 3=5
0 0 0 0 1=2 0 0
0 0 0 0 0 2=3 0
3777777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
45/77
1 2
4 3
1 4
2 5
3
1 2 5
3 4
1 4
2 5 6
3
1 32
4 65
3
4 5
1 2
i j
i i
j j
f1; 2; 3g
P D
24 0 1=2 01 0 1
0 1=2 0
35
2
A
D
P D ADA1
Pn
24 1=4 1=4 1=41=2 1=2 1=2
1=4 1=4 1=4
35
C .1/n
24 1=4 1=4 1=41=2 1=2 1=2
1=4 1=4 1=4
35
8/10/2019 Linear and Its Applications - Lay: Chapter 10
46/77
P
P D24 0 p 0
1 0 10 1 p 0
350 < p < 1
P
E D
266664
1 0 0 0 0
0 1 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 1 0 0
377775
E
EP
P
PE T
P
EP ET
P
A n n E n n In n n
E
EA A
In
E
AT AET
A
In E
EAET A
In E
1 3 2 4
1 3
2 4 f1; 3g f2; 4g
f1; 3g f1; 3g
f2; 4g f2; 4g
P !
266664
1=4
0
3=4
0
1=2
0
1=2
0
1=3
1=3
0
1=3
0
1=3
1=3
1=3
377775
!
266664
1=4
3=4
0
0
1=2
1=2
0
0
1=3
0
1=3
1=3
0
1=3
1=3
1=3
377775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
47/77
i j j
P D
R S
O Q
S P
Q
Q
k!1
Qk D O
Q
j i
j vij i
E vij vij
I 1
0
I D
(0
1
EI D 0 P .I D 0/ C 1 P .I D 1/ D P .I D 1/ D P . /
i j Ik
i k
vij D I0 C I1 C I2 C : : : D
1XkD0
Ik
i 1
vij vij
Ev ij D E
"1XkD0
Ik
#D
1XkD0
EIk D
1XkD0
P .Ik D 1/ D
1XkD0
P . i k/
P . i k/ .i;j / Qk
Evij D
1XkD0
.Qk /ij
i j
. i ; j/
IC Q C Q2 C Q3 C : : : D
1XkD0
Qk
8/10/2019 Linear and Its Applications - Lay: Chapter 10
48/77
IC Q C Q2 C Q3 C : : : D .I Q/1
.I Q/1
M M
j i Q
j .i;j/ M D .I Q/1
i
j j
M D .I Q/1
f1; 2; 3; 4; 5g
3 2
23 4
15234
26666664
1
0
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
0
1
37777775
!
26666664
1
0
0
0
0
0
0
0
0
1
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
37777775
!
26666664
1
0
0
0
0
0
1
0
0
0
1=2
0
0
1=2
0
0
0
1=2
0
1=2
0
1=2
0
1=2
0
37777775
Q M D .I Q/1
Q D
2640
1=2
0
1=2
0
1=2
0
1=2
0
375
M D
2643=2
1
1=2
1
2
1
1=2
1
3=2
375
3 2
M 2 3
1 2
M 2 4
2 4
3
8/10/2019 Linear and Its Applications - Lay: Chapter 10
49/77
tj i
j i i j
tjj j
tj i j i
tj i
j i
P i
pi i S
X Q
SX Q
pi iX
1
O
O
i i
1 S
O Q
tj i i j
M
j
f1; 2; 3; 4; 5g
tj 4 4
j 4
P D
266666640 1=2 0 0 01 0 1=2 0 0
0 1=2 0 1=2 0
0 0 1=2 0 1
0 0 0 1=2 0
37777775 4 4
26666664
0
1
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
1
0
37777775
!
26666664
0
0
1=2
0
1=2
0
1
0
0
0
1=2
0
1=2
0
0
0
1=2
0
1=2
0
0
0
0
1
0
37777775
!
26666664
0
0
0
1=2
1=2
0
0
1
0
0
0
1=2
0
1=2
0
1=2
0
1=2
0
0
1
0
0
0
0
37777775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
50/77
!
2
6666664
1
0
0
0
0
0
0
1
0
0
0
1=2
0
1=2
0
1=2
0
1=2
0
0
1
0
0
0
0
3
7777775
Q M D .I Q/1
Q D
266664
0
1
0
0
1=2
0
1=2
0
0
1=2
0
0
0
0
0
0
377775
M D
266664
3
4
2
0
2
4
2
0
1
2
2
0
0
0
0
1
377775
Mt14 D 9t24D 8t34D 5 t54 D 1
j j
i pi i D 1
P D
I SO Q
j i
j i
. i; j / Pk
j i k i
i i
k
i k .i;j / Pk
j i .i;j / k!1
Pk
Pk
P2 D
I SC SQO Q2
; P3 D
I SC SQ C SQ2O Q3
Pk D
I SkO Qk
Sk D SC SQ C SQ2 C : : : C SQk1
D S.I C Q C Q2 C : : : C Qk1/
8/10/2019 Linear and Its Applications - Lay: Chapter 10
51/77
j i Sk
j i
A D
k!1
Sk D
k!1
S.IC Q C Q2 C : : : C Qk1/
D S.IC Q C Q2 C : : : / D SM
M
. i; j / A j
i
j i
j
i . i; j / A D SM M
S
f1; 2; 3; 4; 5g
1 4
f1;5;2;3;4g
2
6666664
1
0
0
0
0
0
1
0
0
0
1=2
0
0
1=2
0
0
0
1=2
0
1=2
0
1=2
0
1=2
0
3
7777775
Q M D .I Q/1
Q D
264
0
1=2
0
1=2
0
1=2
0
1=2
0
375
M D
264
3=2
1
1=2
1
2
1
1=2
1
3=2
375
A D SM D"
1=2 0 00 0 1=2
#2643=2 1 1=2
1 2 1
1=2 1 3=2
375 D "
3=41=4
1=21=2
1=43=4
#
A 23 4
1 5
4 11=4
8/10/2019 Linear and Its Applications - Lay: Chapter 10
52/77
1
342
5 6
7
1 4 7
4 7
1
4712356 47 2
666666666664
0
1=2
1=2
0
0
0
0
1=3
0
1=3
0
1=3
0
0
1=4
1=4
0
1=4
0
1=4
0
0
0
1
0
0
0
0
0
1=2
0
0
0
1=2
0
0
0
1=3
0
1=3
0
1=3
0
0
0
0
0
1
0
3777777777775
!
2666666666664
00
1
0
0
0
0
00
0
0
0
1
0
01=2
1=2
0
0
0
0
1=30
1=3
0
1=3
0
0
1=41=4
0
1=4
0
1=4
0
01=2
0
0
0
1=2
0
00
1=3
0
1=3
0
1=3
3777777777775
!
2
666666666664
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1=2
1=2
0
0
0
0
1=3
0
1=3
1=3
0
1=4
0
1=4
1=4
0
0
1=4
0
0
0
1=2
0
0
1=2
0
1=3
0
0
1=3
1=3
0
3
777777777775
!
2666666666664
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1=2
1=2
0
0
0
0
1=3
0
1=3
1=3
0
1=4
0
1=4
1=4
0
0
1=4
0
0
0
1=2
0
0
1=2
0
1=3
0
0
1=3
1=3
0
3777777777775
I S
O Q
SD
"0
0
0
0
1=4
0
0
0
0
1=3
#
8/10/2019 Linear and Its Applications - Lay: Chapter 10
53/77
Q D
26666664
0
1=2
1=20
0
1=3
0
1=31=3
0
1=4
1=4
00
1=4
0
1=2
00
1=2
0
0
1=31=3
0
37777775
M D .I Q/1 D
26666664
12=5
12=5
12=5
6=5
6=5
8=5
31=10
34=15
22=15
13=10
6=5
17=10
38=15
14=15
11=10
6=5
11=5
28=15
34=15
8=5
4=5
13=10
22=15
16=15
19=10
37777775
A D SM D
"3=5
2=5
17=30
13=30
19=30
11=30
7=15
8=15
11=30
19=30
#
A 1 4
7 4 73=5
f1;2;3;4g
P D
266664
1 1=2 0 0
0 1=6 1=2 0
0 1=3 1=6 0
0 0 1=3 1
377775
2
2
1
f1;2;3;4g
P D
2666642=3 1=2 0 0
1=3 1=6 1=2 0
0 1=3 1=6 1=2
0 0 1=3 1=2
377775 2
4
2 1
4
8/10/2019 Linear and Its Applications - Lay: Chapter 10
54/77
f1 ; 2 ; : : : ; ng
266664
1 0 1=6 0
0 1 0 1=3
0 0 1=3 2=3
0 0 1=2 0
377775
26666664
1 0 0 1=4 1=5
0 1 0 1=8 1=10
0 0 1 1=8 1=5
0 0 0 1=4 3=10
0 0 0 1=4 1=5
37777775
26666664
1=5 0 1=10 0 1=5
1=5 1 1=5 0 1=5
1=5 0 1=5 0 1=4
1=5 0 1=4 1 1=10
1=5 0 1=4 0 1=4
37777775
A D n!1 Sn
f1 ; 2 ; : : : ; ng
266664
1 0 1=6 0
0 1 0 1=3
0 0 1=3 2=3
0 0 1=2 0
377775
26666664
1 0 0 1=4 1=5
0 1 0 1=8 1=10
0 0 1 1=8 1=5
0 0 0 1=4 3=10
0 0 0 1=4 1=5
37777775
26666664
1=5 0 1=10 0 1=5
1=5 1 1=5 0 1=5
1=5 0 1=5 0 1=4
1=5 0 1=4 1 1=10
1=5 0 1=4 0 1=4
37777775
3
4
4
1
3
1
4
12
3
5 2
4
1 2
4 3
5
5
2
1
5
1
1
5
2
1 2
34
3
2
1
3
1
1
3
2
8/10/2019 Linear and Its Applications - Lay: Chapter 10
55/77
1 2
3 4
1 4
2 5
3
3
4 5
1 2
1 2 3
54
1 32
4 65
3
4 5
1 2
. i; j / M
j
i
.j; i/
i
j i
j .j; i/
A D SM M
S
j M
A m m
An 0n
8/10/2019 Linear and Its Applications - Lay: Chapter 10
56/77
1 2 5
3 4
1 4
2 5 6
3
p D :6
p
p D :7 q
q D :6
p D :7 q D :6
f1; 2; 3; 4; 5g
P D
26666664
1=4 1=2 1=3 0 1=4
3=4 1=2 0 1=3 1=4
0 0 0 1=3 0
0 0 1=3 0 0
0 0 1=3 1=3 1=2
37777775
n!1
Pn
n!1
Pn
P
f1; 2; 3; 4; 5; 6g
P D
26666666664
1=3 1=2 0 0 1=2 0
2=3 1=2 0 0 0 0
0 0 1=4 2=3 0 1=2
0 0 3=4 1=3 0 0
0 0 0 0 1=4 1=4
0 0 0 0 1=4 1=4
37777777775
n!1
Pn
n!1
Pn
P
P D
I S
O Q
Pk D
I SkO Qk
Sk D SC SQ C SQ2 C : : : C SQk1
D S.IC Q C Q2 C : : : C Qk1/:
8/10/2019 Linear and Its Applications - Lay: Chapter 10
57/77
1 4 f1; 4; 2; 3g
P D
266664
1
0
0
0
0
1
0
0
1=2
0
1=6
1=3
0
1=3
1=2
1=6
377775
Q D
"1=6
1=3
1=2
1=6
#
M D
"30=19
12=19
18=19
30=19
#
2
M 2
30
19C
12
19 D
42
19
S D
"1=2
0
0
1=3
#
A D SM D
"15=19
4=19
9=19
10=19
#
1
2 A 1
2 15=19 f4; 1; 2; 3g 4
P D
266664
1
0
0
0
0
2=3
1=3
0
0
1=2
1=6
1=3
1=3
0
1=2
1=6
377775
Q D
264
2=3
1=3
0
1=2
1=6
1=3
0
1=2
1=6
375
M D
264
14:25
7:50
3:00
11:25
7:50
3:00
6:75
4:50
3:00
375
4 2
M 2
11:25 C 7:50 C 3:00 D 21:75
8/10/2019 Linear and Its Applications - Lay: Chapter 10
58/77
14 f1; 4; 2; 3g
P D
2666641
0
0
0
0
1
0
0
1=2
0
1=6
1=3
0
1=3
1=2
1=6
377775
Q D
"1=6
1=3
1=2
1=6
#
; M D
"30=19
12=19
18=19
30=19
#
;
S D
"1=2
0
0
1=3
#
; A D SM D
"15=19
4=19
9=19
10=19
#
2 1 4
A 1
2 15=19
4
8/10/2019 Linear and Its Applications - Lay: Chapter 10
59/77
5
8/10/2019 Linear and Its Applications - Lay: Chapter 10
60/77
28 28
P D
I4 S
O Q
I4 4 4
S 4 24
Q 24 24 S Q
S
24 0 1 2 3 2
pO S
4 8
S D
O O X
XD
2664
pO0
0
0
0
pO0
0
0
pO0
0
0
pO0
0
0
0
pO0
0
0
pO0
0
0
pO0
0
0
0
pO
3775
X 2
3 2 3 4 X
Q 8 8
Q D
24AB
O
O
A
B
O
O
A
35
Q
Q 10 20
1 0 2 A
A B
8 8AB
pWp1p2p3pH
pO pO
S
6
8/10/2019 Linear and Its Applications - Lay: Chapter 10
61/77
8 8 B
B D pO I
I 8 8
A
A
A k 01
2
A D
k k k k k k k k266666666664
pHpW C p1
p2p30
0
0
0
pH0
0
p3pW C p1
0
p20
pH:5p1p2p3
pW:5p1
0
0
pHp1p2p30
pW0
0
pH0
0
p3:5p1
0
p2pW C :5p1
pH0
0
p3p10
p2pW
pH:5p1p2p30
:5p10
pW
pH0
0
p3:5p1
0
p2pW C :5p1
377777777775
k
k
k
k
k
k
k
k
p1p2
p1 C p2
7
8/10/2019 Linear and Its Applications - Lay: Chapter 10
62/77
A
pH
pW C p1
p2 p3
A k
k .1; 3/A k
.1; 3/ pH.2;3/ k
:5
.2;3/ :5p1
.3;3/ k
p2 .3; 3/
.4;3/ k .3; 3/
.4;3/ p3
.5; 3/ k
.5; 3/ pW
.6; 3/ k
p1
:5
.6; 3/ :5p1
.7; 3/ k
.7; 3/
.8; 3/
.8; 3/
8
8/10/2019 Linear and Its Applications - Lay: Chapter 10
63/77
6107
pW D 612=6107 D :1002
p1 D 959=6107 D
:1570 p2 D :0458p3 D :0041pH D :0269 pO D
:660
D 3
R B L
R D B L 3
ER
ER D EB EL 3
EB
P D
I4 S
O Q
1 M D .I Q/1
EB
EL
EL D 0 P .L D 0/ C 1 P .L D 1/ C 2 P .L D 2/ C 3 P .L D 3/
8/10/2019 Linear and Its Applications - Lay: Chapter 10
64/77
8/10/2019 Linear and Its Applications - Lay: Chapter 10
65/77
MS M: M4:6052
4:6052
EB D 4:6052 SM2664:2844:3161
:2725
:1270
3775
EL D 0.:2844/ C 1.:3161/ C 2.:2725/ C 3.:1270/ D 1:2421
ER D EB EL 3 D 4:6052 1:2421 3 D :3631
:3631 9 D 3:2679
10
A .3;6/
.6; 3/A:5p1:
A
M 4:53933
SM 2664
:34973
:33414
:23820
:07793
3775
15;847 29;600 122;268
8/10/2019 Linear and Its Applications - Lay: Chapter 10
66/77
M
S M
M SM
2664
:281776
:292658
:258525
:167041
3775
2664
:268150
:295908
:268120
:167822
3775
2664
:233655
:276714:290207
:199425
3775
M S M
M
SM
M
SM
M
4:53933 S M
2664
:06107
:35881
:41638
:16374
3775
M
4:53933 SM 2664
:06107
:47084
:34791
:12018
3775
M
3:02622 SM
2664:48513
:31279
:16060
:04148
3775
p D :8
p
p
8/10/2019 Linear and Its Applications - Lay: Chapter 10
67/77
.3; 6/A k k
.3; 6/
.6; 3/A k k
.6; 3/ :5p1
8/10/2019 Linear and Its Applications - Lay: Chapter 10
68/77
8/10/2019 Linear and Its Applications - Lay: Chapter 10
69/77
1
3 D
:556
:444
109=216
13=36
:53125
0
5=8
1=8
PD
266664
0 1=3 0 1=2 1=2
1=3 0 1=2 0 1=2
0 1=3 0 1=2 0
1=3 0 1=2 0 0
1=3 1=3 0 0 0
377775
PD
2664
0 0 1 0
1=3 0 0 1=2
1=3 0 0 1=2
1=3 1 0 0
3775
PD
266664
0 1=3 1=4 1=3 0
1=3 0 1=4 0 1=3
1=3 1=3 0 1=3 1=3
1=3 0 1=4 0 1=3
0 1=3 1=4 1=3 0
377775
0 D
266664
0
1
00
0
377775
3 D
266664
:25926
:11111
:25926
:11111
:25926
377775
PD
26666664
0 1=3 0 0 0 0
1=2 0 1=2 0 1=3 0
0 1=3 0 0 0 01=2 0 0 0 1=3 0
0 1=3 0 1 0 0
0 0 1=2 0 1=3 1
37777775
0 D
26666664
1
0
0
0
0
0
37777775
4 D
26666664
:12963
0
:12963
0:43518
:30556
37777775
P
:406
:145375 :448625
3 D2664
1=6
5=185=18
5=18
3775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
70/77
PD
2664
p2 p.1 p/ p.1 p/ .1 p/2
p.1 p/ p2 .1 p/2 p.1 p/
p.1 p/ .1 p/2 p2 p.1 p/
.1 p/2 p.1 p/ p.1 p/ p2
3775 :94206
PD
2666666664
1 p p=6 0 0 0 0 0
p 1 p p=3 0 0 0 0
0 5p=6 1 p p=2 0 0 0
0 0 2p=3 1 p 2p=3 0 0
0 0 0 p=2 1 p 5p=6 0
0 0 0 0 p=3 1 p p
0 0 0 0 0 p=6 1 p
3777777775 :19290
PD
266664
0 1 p p 0 0
p 0 0 0 0
1 p 0 0 0 0
0 p 0 1 0
0 0 1 p 0 1
377775 :192
P m m p
n D 1 n B D Pn
PnC1 D BP . i; j / PnC1 mX
kD1
bik pkj bik p mX
kD1
bik pkj pmX
kD1
pkj
mXkD1
pkj D 1P PnC1 p
P10 D
:33333 :33333
:66667 :66667
D
:33333
:66667
:33333
P20 D
24 :21429 :21429 :21429:57143 :57143 :57143
:21429 :21429 :21429
35
D
24 3=144=7
3=14
35
24 :21429:57143
:21429
35 :21429
8=17 8=17
9=17 9=17
P P2
PD
266664
0 1=4 0 0 0
1 0 1=2 0 0
0 3=4 0 3=4 0
0 0 1=2 0 1
0 0 0 1=4 0
377775
.0;0/ Pk k
.0; 1/ Pk k P
D
266664
1=16
1=4
3=8
1=4
1=16
377775
2
2
PD
2664
0 1=2 0 0
1 0 1=2 0
0 1=2 0 1
0 0 1=2 0
3775
.1;1/ Pk k
.1; 2/ Pk k P
D
26641=6
1=3
1=3
1=6
3775
23
D
266664
1=4
1=4
1=6
1=6
1=6
377775 D
2664
3=13
3=13
3=13
4=13
3775
8/10/2019 Linear and Its Applications - Lay: Chapter 10
71/77
D
266664
:1875
:1875
:25
:1875
:1875
377775 :25
P D
26666664
0 1=3 0 0 0 0
1=2 0 1=2 0 1=3 0
0 1=3 0 0 0 0
1=2 0 0 0 1=3 0
0 1=3 0 1 0 0
0 0 1=2 0 1=3 1
37777775
26666664
0
0
0
0
0
1
37777775D
26666664
0
0
0
0
0
1
37777775
;
152 52 161
G D
266664
:03 :03 :88 :03 :2
:313333 :03 :03 :455 :2
:313333 :03 :03 :455 :2
:313333 :455 :03 :03 :2
:03 :455 :03 :03 :2
377775 ; D
266664
:231535
:208517
:208517
:208517
:142915
377775
.1=2/.1=2/ D 1=4
.1=2/.1=2/ D 1=4
1 1=4 1=4 D 1=2
P6
D
2666666664
1=64
3=32
15=64
5=16
15=64
3=32
1=64
3777777775
3 3
5=16
P D
2666666664
1 p p=6 0 0 0 0 0
p 1 p p=3 0 0 0 0
0 5p=6 1 p p=2 0 0 0
0 0 2p=3 1 p 2p=3 0 0
0 0 0 p=2 1 p 5p=6 0
0 0 0 0 p=3 1 p p
0 0 0 0 0 p=6 1 p
3777777775
2666666664
1=64
3=32
15=64
5=16
15=64
3=32
1=64
3777777775D
2666666664
1=64
3=32
15=64
5=16
15=64
3=32
1=64
3777777775D
p
8/10/2019 Linear and Its Applications - Lay: Chapter 10
72/77
266664
1 1=2 0 0 0
0 0 1=2 0 0
0 1=2 0 1=2 0
0 0 1=2 0 0
0 0 0 1=2 1
377775
266664
q
0
0
0
1 q
377775D
266664
q
0
0
0
1 q
377775
266664
1=4 1=3 1=2 0 0
1=4 1=3 1=4 0 0
1=2 1=3 1=4 0 0
0 0 0 1=3 3=4
0 0 0 2=3 1=4
377775
266664
4=11
3=11
4=11
0
0
377775 D
266664
4=11
3=11
4=11
0
0
377775
266664
1=4 1=3 1=2 0 0
1=4 1=3 1=4 0 0
1=2 1=3 1=4 0 0
0 0 0 1=3 3=4
0 0 0 2=3 1=4
377775
266664
0
0
0
9=17
8=17
377775D
266664
0
0
0
9=17
8=17
377775
12 3 3=5
45 2=5
3
5
266664
4=11
3=11
4=11
0
0
377775C
2
5
266664
0
0
0
9=17
8=17
377775 D
266664
12=55
9=55
12=55
18=85
16=85
377775
12=55
P p C q 1
p 1
q 1
P24 1=31=3
1=3
35
P D 1
P D
jy1j C : : : C jymj jv1j C : : : C jvmj
P D D
jy1j C : : : C jymj D jv1j C : : : C jvmj
1
v1 C : : : C vm
P
D 1
1 P
0 D c1 C c22 C : : : C cnn1 D 1
k D Pk
0 D c1 C c2k2 2 C : : : C cn
knn
k c1 D c2k2 2 C : : : C cn
knn
k ! c1 ji j < 1 j2j
c2k2 2
fkg c1
f1; 3g f2g
f1g f2g f3g
f1;3;5g f2 ; 4 ; 6g
f1; 2; 3; 4; 5g f6g
f1; 2; 3; 4g f5g
f1; 2; 3; 4g
4
15=2
j
i i j
2:27368
G
4:31901
8=3
16=5 D 3:2
8/10/2019 Linear and Its Applications - Lay: Chapter 10
73/77
f g f g f
g
f1; 3g f2g f1; 3g
f2g
1
f1g f2g f3g f1g
f2g f3g
1
f1; 3; 5g f2 ; 4 ; 6g
2
f1; 2; 3; 4g
4
f1; 2; 3; 4g
f5g
1
2; 1; 3 24 1 1=2 1=30 1=4 1=3
0 1=4 1=3
35
1; 3; 5; 2; 4; 6
26666664
0 :4 :8 0 0 0:3 0 :2 0 0 0
:7 :6 0 0 0 0
0 0 0 0 :7 :5
0 0 0 :1 0 :5
0 0 0 :9 :3 0
37777775
266664
1=3 0 1 0 0
1=3 0 0 1=2 0
1=3 0 0 1=2 0
0 1=2 0 0 0
0 1=2 0 0 1
377775
5; 1; 2; 3; 4 266664
1 0 1=2 0 00 1=3 0 1 0
0 1=3 0 0 1=2
0 1=3 0 0 1=2
0 0 1=2 0 0
377775
f1; 2; 3; 4; 5g f6g
f1; 2; 3; 4; 5g f6g
26666664
0 1=3 0 0 0 0
1=2 0 1=2 0 1=3 0
0 1=3 0 0 0 0
1=2 0 0 0 1=3 0
0 1=3 0 1 0 0
0 0 1=2 0 1=3 1
37777775
6; 1; 2; 3; 4; 5 26666664
1 0 0 1=2 0 1=3
0 0 1=3 0 0 0
0 1=2 0 1=2 0 1=3
0 0 1=3 0 0 0
0 1=2 0 0 0 1=3
0 0 1=3 0 1 0
37777775
D
2664
1=4
1=4
1=4
1=4
3775 P
P2 D
2664
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
3775; P3 D
2664
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
3775;
P4 D2664
1 0 0 0
0 1 0 00 0 1 0
0 0 0 1
3775 D I
P5 D PP6 D P2P7 D P3
P8 D P4 D I n
PnC1PnC2PnC3 PnC4
P P2 P3 P4 D I
n!1
1
4
PnC1 C PnC2 C PnC3 C PnC4
D 1
4 PC P2 C P3 C P4
D 1
4
0BB@2664
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
3775C
2664
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
3775
C
2664
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
3775C
2664
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
37751CCA
8/10/2019 Linear and Its Applications - Lay: Chapter 10
74/77
D
2664
1=4 1=4 1=4 1=4
1=4 1=4 1=4 1=4
1=4 1=4 1=4 1=4
1=4 1=4 1=4 1=4
3775
2
D
24 1=41=2
1=4
35
D D
24 1 0 00 1 0
0 0 0
35 ; A D
24 1 1 12 2 0
1 1 1
35
P
n
D AD
n
A
1
D24
1 1 1
2 2 01 1 13524
1 0 0
0 .1/
n
00 0 03524
1=4 1=4 1=4
1=4 1=4 1=41=2 0 1=235
D 1
4
24 1 C .1/
n 1 .1/n 1 C .1/n
2 2.1/n 2C 2.1/n 2 2.1/n
1 C .1/n 1 .1/n 1 C .1/n
35
D
24 1=4 1=4 1=41=2 1=2 1=2
1=4 1=4 1=4
35C .1/n
24 1=4 1=4 1=41=2 1=2 1=2
1=4 1=4 1=4
35
Pn PnC1
.1=2/.Pn C PnC1/ D
2
41=4 1=4 1=4
1=2 1=2 1=2
1=4 1=4 1=4
3
5
D
24 1=31=3
1=3
35 P
P2 D
24 0 1 00 0 1
1 0 0
35 P3 D
24 1 0 00 1 0
0 0 1
35 D I
P4 D PP5 D P2P6 D P3 D I n
PnC1PnC2 PnC3 P P2 P3 D I
n!1
1
3
PnC1 C PnC2 C PnC3
D
1
3
PC P2 C P3
D 1
3
0@24 0 0 11 0 0
0 1 0
35C
24 0 1 00 0 1
1 0 0
35C
24 1 0 00 1 0
0 0 1
351A
D
24 1=3 1=3 1=31=3 1=3 1=3
1=3 1=3 1=3
35
8/10/2019 Linear and Its Applications - Lay: Chapter 10
75/77
A
E1 : : : Ek
E D Ek E1 EA A
In E
EAT AT
In E .EAT/T
A
In E
.EAT/T D .AT/TET D AET
EAE T .EA/ET EA
A
In E
EA.EA/ET EA
In E
EAET A
In E
.EA/ET D E.AET/
A A
3 2
3=2 2
24 1075=736 125=368 185=36825=46 35=23 15=23
105=184 55=92 155=92
35
24 10=21 3=75=21 3=14
2=7 5=14
35
3=2 1895=736
10=21 5=21 2 =7
9=11 29=11 3=7
1 10=3
5=7 38=5
. i; j / M
i
j
2=7 19=2 3:84615
2:53846
3:30769
3:01105 3:92932
3:92932 3:01105 D :91827
f1; 2g f3; 4; 5g f1; 2g
2=5 2=5
3=5 3=5
f1; 2g 1
1 2=5
2
3=5 34 5
0
i n!1
Pn
i
n!1
Pn D
266664
2=5 2=5 2=5 2=5 2=5
3=5 3=5 3=5 3=5 3=5
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
377775
P50
266664
:4 :4 :4 :4 :4
:6 :6 :6 :6 :6
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
377775
n D 1
n D k Pk D
I SkO Qk
Sk D S
IC Q C Q2 C : : : C Qk1
PkC1 D Pk P
D
I SkO Qk
I S
O Q
D
I SC Sk Q
O QkC1
SC Sk Q D SC S.IC Q C Q2 C : : : C Qk1/Q
D SC S.Q C Q2 C : : : C Qk /
D S.IC Q C Q2 C : : : C Qk /
D SkC1
A
k k .1;2/A
k
.1;2/ pH
8/10/2019 Linear and Its Applications - Lay: Chapter 10
76/77
.2;2/ k
.2;2/ 0
.3;2/ k
.3;2/ 0
.4;2/ k
.4;2/ p3
.5;2/ k
.5;2/ pWC p1
.6;2/ k
.6;2/ 0
.7; 2/ k
.7; 2/ p2
.8;2/ k
.8;2/ 0
A k
k
.1;5/A k
.1;5/ pH
.2;5/ k
.2;5/ 0
.3;5/ k
.3;5/
0
.4;5/ k
.4;5/ p3
.5;5/ k
:5 .5;5/ :5p1
.6;5/ k
.6;5/ 0
.7; 5/ k
.7; 5/ p2
.8;5/ k
.8;5/ pWC :5p1
A
k k
.1;7/A k
.1;7/ pH
.2;7/ k
:5 .2;7/ :5p1
.3;7/ k
p2 .3;7/
.4;7/ k .3;6/ .4;7/ p3
.5;7/ k
.5;7/ 0
.6;7/ k
p1
:5
.6;7/
:5p1.7; 7/ k
.7; 7/
.8;7/ k
.8;7/ pW
8/10/2019 Linear and Its Applications - Lay: Chapter 10
77/77
pWD :0954785p1 D :159996p2 D :049377p3 D :00514581pHD :0291127pO D :66089
A D
k k k k k k k k2
66666666664
:0291127
:255474
:049377
:00514581
0
0
0
0
:0291127
0
0
:00514581
:255474
0
:049377
0
:0291127
:0799978
:049377
:00514581
:0954785
:0799978
0
0
:0291127
:159996
:049377
:00514581
0
:0954785
0
0
:0291127
0
0
:00514581
:0799978
0
:049377
:175476
:0291127
0
0
:00514581
:159996
0
:049377
:0954785
:0291127
:0799978
:049377
:00514581
0
:0799978
0
:0954785
:0291127
0
0
:00514581
:0799978
0
:049377
:175476
3
77777777775
k
k
k
k
k
k
k
k
M
EB D 4:53933 SMEL
EL D 0.:34973/ C 1.:33414/ C 2.:23820/ C 3.:07793/
D 1:04433
ER D EB EL 3D 4:53933 1:04433 3 D :495
pWD :212933p1D :119495p2 D :0480377
p3 D :00615458pHD :0609064pO D :552474
pWD :2004p1D :144421p2 D :0481721
p3 D :0129474pHD :0679741pO D :526085
pWD :210936p1 D :157383p2 D :0537579
p3 D :00727012pHD :0533484pO D :517305
M
SM
M 4:53933
EB D 1 C 4:53933 D 5:53933 SMEL
EL D 0.:06107/ C 1.:47084/ C 2.:34791/ C 3.:12108/
D 1:52990
ER D EB EL 3
D 5:53933 1:52990 3 D 1:00943
:85654
1:00943
:26779
1:00943.:8/ C :26779.:2/ D :861102
