1. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Grammik 'Algebra
Apaloif tou Gkˆouc
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
29 SeptembrÐou 2014

2. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Eidik¸n Susthmˆtwn (m = n)
Me diag¸niouc pÐnakec suntelest¸n: ai;j = 0; 8i6= j
a1;1x1 + 0x2 + : : : + 0xn = b1
0x1 + a2;2x2 + : : : + 0xn = b2
...
0x1 + 0x2 + : : : + an;nxn = bn
9>>>>= >>>>;

3. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Eidik¸n Susthmˆtwn (m = n)
Me diag¸niouc pÐnakec suntelest¸n: ai;j = 0; 8i6= j
a1;1x1 + 0x2 + : : : + 0xn = b1
0x1 + a2;2x2 + : : : + 0xn = b2
...
0x1 + 0x2 + : : : + an;nxn = bn
9>>>>= >>>>;
) xj = bj=aj;j; 1 j n
Me trigwnikoÔc pÐnakec suntelest¸n: ai;j = 0; 8i j
a1;1x1 + 0x2 + : : : + 0xn = b1
a2;1x1 + a2;2x2 + : : : + 0xn = b2
...
an;1x1 + an;2x2 + : : : + an;nxn = bn
9=
;

4. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Eidik¸n Susthmˆtwn (m = n)
Me diag¸niouc pÐnakec suntelest¸n: ai;j = 0; 8i6= j
a1;1x1 + 0x2 + : : : + 0xn = b1
0x1 + a2;2x2 + : : : + 0xn = b2
...
0x1 + 0x2 + : : : + an;nxn = bn
9= ;
) xj = bj=aj;j; 1 j n
Me trigwnikoÔc pÐnakec suntelest¸n: ai;j = 0; 8i j
a1;1x1 + 0x2 + : : : + 0xn = b1
a2;1x1 + a2;2x2 + : : : + 0xn = b2
...
an;1x1 + an;2x2 + : : : + an;nxn = bn
9=
;
)
xj =
0
@bj
Xj1
k=1
aj;kxk
1
A=aj;j; 1 j n

5. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
Epauxhmènoc PÐnakac
2
666664
1 2 2 6 1
0 1 5 6 15
0 0 1 17
5 4
0 0 0 1 0
3
777775

6. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
Epauxhmènoc PÐnakac
2
666664
1 2 2 6 1
0 1 5 6 15
0 0 1 17
5 4
0 0 0 1 0
3
777775
TeleutaÐa exÐswsh x4 = 0.
3h exÐswsh x3 17
5 x4 = 4 ) x3 = 4.
2h exÐswsh x2 + 5x3 6x4 = 15 ) x2 = 5.
1h exÐswsh x1 + 2x2 + 2x3 6x4 = 1 ) x1 = 3.

7. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
Epauxhmènoc PÐnakac
2
666664
1 2 2 6 1
0 1 5 6 15
0 0 1 17
5 4
0 0 0 1 0
3
777775
TeleutaÐa exÐswsh x4 = 0.
3h exÐswsh x3 17
5 x4 = 4 ) x3 = 4.
2h exÐswsh x2 + 5x3 6x4 = 15 ) x2 = 5.
1h exÐswsh x1 + 2x2 + 2x3 6x4 = 1 ) x1 = 3.
LÔsh: x1 = 3; x2 = 5; x3 = 4; x4 = 0.

8. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Je¸rhma Ôparxhc kai monadikìthtac lÔshc
Kˆje trigwnikì sÔsthma èqei
I monadik lÔsh ann aj;j6= 0 8j
I kammÐa lÔsh ann gia kˆpoio j, ajj = 0 kai
bj
Pj1
k=1 aj;kxk6= 0
I ˆpeirec jPlÔseic ann gia kˆpoio , ajj = 0 kai
bj j1
k=1 aj;kxk = 0

9. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Genik¸n Susthmˆtwn (m = n)
Gia na lÔsw (melet sw) èna genikì sÔsthma
I to metatrèpw se isodÔnamo trigwnikì
I lÔnw (melet¸) to trigwnikì

10. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Genik¸n Susthmˆtwn (m = n)
Gia na lÔsw (melet sw) èna genikì sÔsthma
I to metatrèpw se isodÔnamo trigwnikì
I lÔnw (melet¸) to trigwnikì
DÔo sust mata exis¸sewn eÐnai isodÔnama ann ìlec oi lÔseic tou
enìc eÐnai kai lÔseic tou ˆllou.

11. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Genik¸n Susthmˆtwn (m = n)
Gia na lÔsw (melet sw) èna genikì sÔsthma
I to metatrèpw se isodÔnamo trigwnikì
I lÔnw (melet¸) to trigwnikì
DÔo sust mata exis¸sewn eÐnai isodÔnama ann ìlec oi lÔseic tou
enìc eÐnai kai lÔseic tou ˆllou.
To sÔnolo twn lÔsewn enìc sust matoc den allˆzei an

12. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Genik¸n Susthmˆtwn (m = n)
Gia na lÔsw (melet sw) èna genikì sÔsthma
I to metatrèpw se isodÔnamo trigwnikì
I lÔnw (melet¸) to trigwnikì
DÔo sust mata exis¸sewn eÐnai isodÔnama ann ìlec oi lÔseic tou
enìc eÐnai kai lÔseic tou ˆllou.
To sÔnolo twn lÔsewn enìc sust matoc den allˆzei an
I pollaplasiˆsw mia exÐsws tou me ènan mh-mhdenikì arijmì

13. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
EpÐlush Genik¸n Susthmˆtwn (m = n)
Gia na lÔsw (melet sw) èna genikì sÔsthma
I to metatrèpw se isodÔnamo trigwnikì
I lÔnw (melet¸) to trigwnikì
DÔo sust mata exis¸sewn eÐnai isodÔnama ann ìlec oi lÔseic tou
enìc eÐnai kai lÔseic tou ˆllou.
To sÔnolo twn lÔsewn enìc sust matoc den allˆzei an
I pollaplasiˆsw mia exÐsws tou me ènan mh-mhdenikì arijmì
I prosjèsw mia exÐswsh se mia ˆllh
I enallˆxw thn seirˆ dÔo exis¸sewn

14. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn

15. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0

16. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0
I Antikatast soume mia
exÐswsh me ton eautì thc
sun to pollaplˆsio miac
ˆllhc exÐswshc

17. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0
I Antikatast soume mia
exÐswsh me ton eautì thc
sun to pollaplˆsio miac
ˆllhc exÐswshc
Prˆxeic:
I Enˆllaxe thn seirˆ dÔo
gramm¸n (enallag )

18. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0
I Antikatast soume mia
exÐswsh me ton eautì thc
sun to pollaplˆsio miac
ˆllhc exÐswshc
Prˆxeic:
I Enˆllaxe thn seirˆ dÔo
gramm¸n (enallag )
I Pollaplasiasmìc mia
gramm c me c6= 0
(stˆjmish)

19. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0
I Antikatast soume mia
exÐswsh me ton eautì thc
sun to pollaplˆsio miac
ˆllhc exÐswshc
Prˆxeic:
I Enˆllaxe thn seirˆ dÔo
gramm¸n (enallag )
I Pollaplasiasmìc mia
gramm c me c6= 0
(stˆjmish)
I Antikatˆstash mia
grammlhc me ton eautì
thc sun to pollaplˆsio
miac ˆllhc gramm c
(Antikatˆstash)

20. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
To sÔnolo twn lÔsewn enìc grammikoÔ sust matoc exis¸sewn
paramènei analoÐwto an:
I Enallˆxoume thn seirˆ
twn exis¸sewn
I Pollaplasiˆsoume kˆpoia
exÐswsh me ènan arijmì
c6= 0
I Antikatast soume mia
exÐswsh me ton eautì thc
sun to pollaplˆsio miac
ˆllhc exÐswshc
Prˆxeic:
I Enˆllaxe thn seirˆ dÔo
gramm¸n (enallag )
I Pollaplasiasmìc mia
gramm c me c6= 0
(stˆjmish)
I Antikatˆstash mia
grammlhc me ton eautì
thc sun to pollaplˆsio
miac ˆllhc gramm c
(Antikatˆstash)
stìqoc: QrhsimopoÐhse tic parapˆnw prˆxeic gia na aplopoi seic
to prìblhma.

21. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
1x1 + 2x2 = 3
2x1 + 1x2 = 3

22. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
1x1 + 2x2 = 3
2x1 + 1x2 = 3
AfaÐrese dÔo forèc thn 1h apo
thn 2h:
1x1 + 2x2 = 3
3x2 = 3

23. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
1x1 + 2x2 = 3
2x1 + 1x2 = 3
AfaÐrese dÔo forèc thn 1h apo
thn 2h:
1x1 + 2x2 = 3
3x2 = 3
1 2 3
2 1 3

24. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Parˆdeigma
1x1 + 2x2 = 3
2x1 + 1x2 = 3
AfaÐrese dÔo forèc thn 1h apo
thn 2h:
1x1 + 2x2 = 3
3x2 = 3
1 2 3
2 1 3
1 2 3
0 3 3

25. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Pr¸to b ma:
QrhsimopoÐhse ton x1 ˆgnwsto [ˆnw aristerˆ] gia na apaloÐyete
ìlouc touc x1 upìloipouc ìrouc [kˆne ta upìloipa stoiqeÐa thc
pr¸thc st lhc 0]:
L1 : x1 3x2 2x3 = 6
L2 : 2x1 4x2 3x3 = 8
L3 : 3x1 + 6x2 + 8x3 = 5

26. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Pr¸to b ma:
QrhsimopoÐhse ton x1 ˆgnwsto [ˆnw aristerˆ] gia na apaloÐyete
ìlouc touc x1 upìloipouc ìrouc [kˆne ta upìloipa stoiqeÐa thc
pr¸thc st lhc 0]:
L1 : x1 3x2 2x3 = 6
L2 : 2x1 4x2 3x3 = 8
L3 : 3x1 + 6x2 + 8x3 = 5
AfaÐrese 2 1h apo thn 2h
Prìsjese 3 1h sthn 3h:
L1 : x1 3x2 2x3 = 6
L2 2L1 : 2x2 + x3 = 4
3L1 + L3 : 3x2 + 2x3 = 13

27. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Pr¸to b ma:
QrhsimopoÐhse ton x1 ˆgnwsto [ˆnw aristerˆ] gia na apaloÐyete
ìlouc touc x1 upìloipouc ìrouc [kˆne ta upìloipa stoiqeÐa thc
pr¸thc st lhc 0]:
L1 : x1 3x2 2x3 = 6
L2 : 2x1 4x2 3x3 = 8
L3 : 3x1 + 6x2 + 8x3 = 5
AfaÐrese 2 1h apo thn 2h
Prìsjese 3 1h sthn 3h:
L1 : x1 3x2 2x3 = 6
L2 2L1 : 2x2 + x3 = 4
3L1 + L3 : 3x2 + 2x3 = 13
2
4
1 3 2 6
2 4 3 8
3 6 8 5
3
5

28. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Pr¸to b ma:
QrhsimopoÐhse ton x1 ˆgnwsto [ˆnw aristerˆ] gia na apaloÐyete
ìlouc touc x1 upìloipouc ìrouc [kˆne ta upìloipa stoiqeÐa thc
pr¸thc st lhc 0]:
L1 : x1 3x2 2x3 = 6
L2 : 2x1 4x2 3x3 = 8
L3 : 3x1 + 6x2 + 8x3 = 5
AfaÐrese 2 1h apo thn 2h
Prìsjese 3 1h sthn 3h:
L1 : x1 3x2 2x3 = 6
L2 2L1 : 2x2 + x3 = 4
3L1 + L3 : 3x2 + 2x3 = 13
2
4
1 3 2 6
2 4 3 8
3 6 8 5
3
5
2
4
1 3 2 6
0 2 1 4
0 3 2 13
3
5
Proq¸rhse sto deÔtero b ma! - deÔterh st lh

29. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
DeÔtero b ma:
L1 : x1 3x2 2x3 = 6
L2 : 2x2 + x3 = 4
L3 : 3x2 + 2x3 = 13

30. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
DeÔtero b ma:
L1 : x1 3x2 2x3 = 6
L2 : 2x2 + x3 = 4
L3 : 3x2 + 2x3 = 13
Prìsjese 32
L2 sthn L3:
L1 : x1 3x2 2x3 = 6
L2 : 2x2 + x3 = 4
L3 +
3
2
L2 :
7
2
x3 = 7

31. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
DeÔtero b ma:
L1 : x1 3x2 2x3 = 6
L2 : 2x2 + x3 = 4
L3 : 3x2 + 2x3 = 13
Prìsjese 32
L2 sthn L3:
L1 : x1 3x2 2x3 = 6
L2 : 2x2 + x3 = 4
L3 +
3
2
L2 :
7
2
x3 = 7
2
4
1 3 2 6
0 2 1 4
0 3 2 13
3
5
2
4
1 0 12
0
72
0 2 1 4
0 0 7
3
5
Pol/sÐase thn L3 me 2
1 0 0
12
27
4
0 2 1 4
0 0 1 2
3
5
OrÐste h lÔsh: (x1; x2; x3) = (1;3; 2)

32. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Ja upˆrxoun probl mata?
x1 + 2x2 = 3
x1 + 2x2 = 4

33. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Ja upˆrxoun probl mata?
x1 + 2x2 = 3
x1 + 2x2 = 4
AfaÐrese thn L1 apo thn L2:
x1 + 2x2 = 3
0= 1
asunèpeia
KAMM'IA L'USH

34. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Ja upˆrxoun probl mata?
x1 + 2x2 = 3
x1 + 2x2 = 4
AfaÐrese thn L1 apo thn L2:
x1 + 2x2 = 3
0= 1
asunèpeia
KAMM'IA L'USH
x1 + 2x2 = 3
2x1 + 4x2 = 6

35. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Ja upˆrxoun probl mata?
x1 + 2x2 = 3
x1 + 2x2 = 4
AfaÐrese thn L1 apo thn L2:
x1 + 2x2 = 3
0= 1
asunèpeia
KAMM'IA L'USH
x1 + 2x2 = 3
2x1 + 4x2 = 6
AfaÐrese 2L1 apo L2:
x1 + 2x2 = 3
0= 0
asˆfeia
'APEIRES L'USEIS

36. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Er¸thsh
Pìsec lÔseic èqei to parakˆtw sÔsthma?
5x1 + 2x2 3x3 = 4
12x1 7x2 + 2x3 = 8
3x1 + 4x2 + 5x3 = 10

37. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
Orismìc
Apaloif tou Gkˆouc eÐnai
ènac algìrijmoc pou
metatrèpei èna sÔsthma se
èna isodÔnamo ˆnw trigwnikì.

38. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
SÔsthma 4 4
x2 + 2 x3 x4 = 1
x1 + x3 + x4 = 4
x1 + x2 x4 = 2
2 x2 + 3 x3 x4 = 7
A =
0
BB@
1
0 1 2 1
1 0 1 1
1 1 0 1
0 2 3 1
CCA
b =
0
BB@
1
1
4
2
7
CCA

39. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
SÔsthma 4 4
M =
0
BB@
0 1 2 1
1 0 1 1
1 1 0 1
0 2 3 1

47. 1427
1
CCA

48. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
SÔsthma 4 4
0
0 1 2 1
1 0 1 1
1 1 0 1
0 2 3 1
BB@

56. 1
4
2
7
1
CCA
enallagèc ex.
!
0
1 0 1 1
1 1 0 1
0 1 2 1
0 2 3 1
BB@

64. 1
4
2
1
7
CCA
E1+E2!E2
!
0
1 0 1 1
0 1 1 0
0 1 2 1
0 2 3 1
BB@

72. 1
4
6
1
7
CCA
E3E2!E3
E42 E2!E4
!
0
BB@
1 0 1 1
0 1 1 0
0 0 1 1
0 0 1 1

80. 4
6
5
5
1
CCA
E4E3!E4
!
0
1 0 1 1
0 1 1 0
0 0 1 1
0 0 0 0
BB@

88. 1
4
6
5
0
CCA

89. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
SÔsthma 4 4
0
0 1 2 1
1 0 1 1
1 1 0 1
0 2 3 1
BB@

97. 1
4
2
7
1
CCA
enallagèc ex.
!
0
1 0 1 1
1 1 0 1
0 1 2 1
0 2 3 1
BB@

105. 1
4
2
1
7
CCA
E1+E2!E2
!
0
1 0 1 1
0 1 1 0
0 1 2 1
0 2 3 1
BB@

113. 1
4
6
1
7
CCA
E3E2!E3
E42 E2!E4
!
0
BB@
1 0 1 1
0 1 1 0
0 0 1 1
0 0 1 1

121. 4
6
5
5
1
CCA
E4E3!E4
!
0
1 0 1 1
0 1 1 0
0 0 1 1
0 0 0 0
BB@

129. 1
4
6
5
0
CCA

130. EpÐlush Trigwnik¸n Susthmˆtwn EpÐlush Genik¸n Susthmˆtwn
H lÔsh
x4 = k,
x3 = 5 + k,
x2 = 11 k
x1 = 9 2k
ìpou k opoiosd pote
pragmatikìc arijmìc.