رياضيات عامة 113 ريض
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−=−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛ −
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
0427.001325
,011726.0,
25.0
01
,12105735
,1210
57
35
EDCBA
W ٢WאאFאאEאKK
٢WאאאאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
×−+=
0427.021325
,06227.001
31223GF
BA ≠،GE ≠אFאאאאE،FE =K
![Page 23: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/23.jpg)
א ١١٣ אא
א אאא
٢KאW ٢}١אאW
٣WFE،אאFEאאאאK
٣WאאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
=71
40
32
136742
BA
W .)2,)1 ABBA −+
אW
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−−−−
−−−−=−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−++−++−
=+
819644
)1(734)6(3714022
)2
673840
714336170422
)1
AB
BA
١WאK ٤WאאאאKאW
.)3),()2,)1 ABBBABAB +++++ אW
1Eא1אאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=+=+673
840BAAB
2Eא1אאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=++=++13110942
71
40
32
673840
)()( BABBAB
3Eא2אאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−=++=++=++
13110942
)( BABBABABB
![Page 24: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/24.jpg)
א ١١٣ אא
א אאא
٢}٢אW ٤WFEFE،א
FEאאאאFאEFEאאK
٥WאאW
.4.02
1
305.0
,520
613
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛= BA
W
.2
)3,2)2,3)1−
+−BBAA
אW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
×××
×××
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛×=
1560
1839
532303
631333
520
613
33)1 A
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−−−
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−×+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛×−=+−
2.46
2
012
8.04
2
601
52
0
613
4.02
1
305.0
2520
613
12)2 BA
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−
−
−=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−−−
−
−
−
=−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
=−
2.01
5.0
5.10
25.0
24.0222
1
232
025.0
24.02
1
305.0
2)3 B
٢WFKE ٦WאאאKאW
)2(5.1)23)1 AA×
![Page 25: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/25.jpg)
א ١١٣ אא
א אאא
אW 1Eא(1)אאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==×
1560
1839
33 AA
2Eא(1)אאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==×=
1560
1839
3)25.1()2(5.1 AAA
٢}٣W ٥Wאא
אאאא،אאאK ٧WאW
( ) ( ) .413
3.0021)2,
10413
3.0021)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
−= ba
אW 23023)103.0()40()1)2(())3(1()1 −=++−−=×+×+×−+−×=a
2EbאאK אאאאאK
٢}٤W ٦Wאkm×אnk ×F
אאאאEאnm×،אאאאאאאאאאK
![Page 26: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/26.jpg)
א ١١٣ אא
א אאא
٨WאאW
.632312015
3021,
2321
11,
30
11
52
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
= CBA
אW .)3,)2,)1 BCBAAB
אW
( ) ( )
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
=315
41
221
31531
1315
221
01231
1012
)1 AB
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−=
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛
=654
638307
30
231
123
52
23
30
211
121
52
21
30
111
111
52
11
)2 BA
3Eאאא2אא3K
Wא(1)(2)אאאW BAAB ≠אK
٣WאK ٩Wא(1)אא٧W
( )3.0021
10413
−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
=A
![Page 27: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/27.jpg)
א ١١٣ אא
א אאא
אW
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
××−××××−××××−×××−×−−×−×−
=
3020102.10843.00219.0063
3.0100102101103.040424143.010121113.03032313
A
W
( ) ( )3.0021
10413
10413
3.0021 −
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
≠
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
−
אאאאאأي n×1א1×nאK
١٠WאאW
( ) .4003
12,5.003,
21
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−==⎟⎟
⎠
⎞⎜⎜⎝
⎛−= CBA
אW ).()2,)()1 BCACAB
אW
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−=
101256
4003
12
1065.003
)()1 CAB
2Eא1אאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−==
101256
)()( CABBCA
٤W،אW
ACABCBABCACCBA
+=++=+
)()2)()1
אABCאאאאK
![Page 28: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/28.jpg)
א ١١٣ אא
א אאא
אW ٧WAאnm×אtAאmn×،
Kאאאא ١١WאאאאאK אW
( ) .401032
,5.0
03
,21 ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=−= ttt CBA
٣KאאW ٣}١אאW
٨WאAאK ١٢WאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
510604
,316
035410
,102124301
,1132
,0111
ED
CBA
C33×وDالمصفوفات A22×وBالمصفوفات אE
٣}٢אW ٩Wאא،1א
אאKnIאאnn×،IאאK
![Page 29: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/29.jpg)
א ١١٣ אא
א אאא
١٣W
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
1000010000100001
100010001
1001
432 III
٥WאאK ١٤WאW
.251430321
100010001
)2,100010001
251430321
)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−= BA
אW
.251430321
251430321
100010001
)2,251430321
100010001
251430321
)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−= BA
٤KאאW ١٠Wאא nn×אאא
אK(A)detK
Kאאא11×K
٤}١אא22×W ١١W22א×⎟⎟
⎠
⎞⎜⎜⎝
⎛=
dcba
AאאFאE
אאFאE،W
bcaddcba
(A) −==det
![Page 30: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/30.jpg)
א ١١٣ אא
א אאא
١٥WאאאאW
0511
)2,3142
)1−
אW
5)5(0)15()01(0511
)2,246)14()32(3142
)1 =−−=−×−×=−
=−=×−×=
٤}٢אא33×W ١٢W33א×אAאאאא
FאEאאאאFאE،אאאאK
א⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
333231
232221
131211
aaaaaaaaa
AW
122133112332132231322113312312332211
3231
2221
1211
333231
232221
131211
333231
232221
131211
det
aaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
(A)
−−−++=
==
١٦WאאאאW
.420315
201)2,
478362931
)1−
−−
אW
25524214321267224)324()137()968()729()833()461(
786231
478362931
478362931
)1
−=−−−++=××−××−××−××+××+××=
=
![Page 31: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/31.jpg)
א ١١٣ אא
א אאא
180)6(02004)054()132()210()252()030()411(
201501
420315
201
420315
201)2
−=−−−−−+−=××−−×−×−−××−−××+×−×+××−=
−
−
−−
−=
−−
−
١٧WאאאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
510604
,316
035410
,102124301
,1132
,0111
ED
CBA
אW
107150722000131
65
0
316035410
316035410
)det(
1000)12(00202
0
241
102124301
102124301
)det(
5321132
)det(,1)1(00111
)det(
−=−−−−+=−−
−=−
−=
=−−−−++−=−−−
−−
=−
−−
=
−=−−=−
==−−=−
=
D
C
BA
)det(EEK ٦WK
١٨WאאאKאאאאW ).det()4),det()3),det()2),det()1 DCCDBAAB
![Page 32: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/32.jpg)
א ١١٣ אא
א אאא
אW אאאאKא
אW
107010107)det()det()det()4107010710)det()det()det()3
515)det()det()det()2551)det()det()det()1
−=×−=×=−=−×=×=
−=×−=×=−=−×=×=
CDDCDCCDABBABAAB
5KאW ١٣WAאא1−A J–
،אW IAAAA == −− 11
אא22×
٧WאAאא 22×W
٨Wאabcdא0det ≠−= bcadAאW
dcbaacbd
bcadacbd
dcba ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
=−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−1
אא22×אK ١٩Wאא22× אW
.3162
,4321
)2,0213
)1 ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −= CBA
אW 1EאAאW
![Page 33: رياضيات عامة 113 ريض](https://reader033.vdocuments.mx/reader033/viewer/2022061314/55cf9a3f550346d033a0fd06/html5/thumbnails/33.jpg)
א ١١٣ אא
א אאא
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
=−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ − −
5.115.00
23210
021332
)1(0
0213 1
2EאBאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
5.05.112
21324
43211324
4321 1
3EאCאK
١WאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−−
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−−=
⎟⎟⎟
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א ١١٣ אא
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אא
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א ١١٣ אא
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א ١١٣ אא
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1065
=x12=x
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0=+ baxabא0≠a
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א ١١٣ אא
א א
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א ١١٣ אא
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א ١١٣ אא
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א ١١٣ אא
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zyx
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א ١١٣ אא
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xx
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א א
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WאW 7,1 == yx
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D −==
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א ١١٣ אא
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אאאא
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א ١١٣ אאא
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אאWאא
אאWאאאאאאאאK
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• אאא • אאאאK • אאאאK • אאאאK • אאK
אאאWאא٨٠٪K אאWK
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א ١١٣ אאא
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0
1
2
3
6
4
2
8
X Y
f
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א ١١٣ אאא
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22
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א ١١٣ אאא
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٤WאאWCitiesCountriesאgCitiesCountriesW)(xgאאאxK
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==
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א ١١٣ אאא
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א ١١٣ אאא
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W2)( xxg = אאfא٧אKgf =؟
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אאNN: →g2)( xxg =אK
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א ١١٣ אאא
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אFאאKE
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אאWאאW
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0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
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א ١١٣ אאא
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0.5 1 1.5 2
0.2
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1.4
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١ER=fDK ٢E}{aR f =אK ٣E)()( xfxf =−K ٤EאאK
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א ١١٣ אאא
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-1
1
2
3
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א ١١٣ אאא
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cאאאK אW
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2
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א ١١٣ אאא
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אאWRR: →f112)(
−+
=xxxf
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א ١١٣ אאא
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א ١١٣ אאא
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١٤٠٣ J١٩٨٣K
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3) Gwyn Davies and Gordon Hick, Mathematics for scientific and technical students,
Addison Wesley Longman, Harlow, England, 1998. 4) Anders Hald, A History of Probability and Statistics and Their Applications before
1750, John Wiley and Sons, New York, 1989.
5) Alexander Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons,
Chichester, England, 1986.
6) Seymour Lipschutz and Marc Lipson, Discrete Mathematics, McGraw-Hill, New York,
1997. 7) Peter Tebbutt, Basic Mathematics, John Wiley & Sons, Chichester, England, 1998.
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GOTEVOT appreciates the financial support provided by BAE SYSTEMS