TEMA:Determinantes y Multiplicación de

Matrices UNIDAD:

IIIMATERIA:

Álgebra LinealDOCENTE:

ALUMNOS:Jorge Carrasco Toledo, Yuset Martínez López, Josué Rafael Gamboa Caballero, Lizbet Andrea Ramírez

Pérez, Lessly Elvira Ocaña Portillo, Mauricio González Niño, Henry Alfonso Hernández Lázaro

INSTITUTO TECNOLÓGICO SUPERIOR DE CINTALAPA.

1.- [312231321]

3 [ (3∗1 )−(1∗2 ) ]−1 [ (2∗1 )−(1∗3 ) ]+2 [ (2∗2 )−(3∗3 ) ]

3 (1 )−1 (−1 )+2 (5 )=3+1−10=−6

2.- [−15 470−3120 ]

−1 [ (0∗6 )−(3∗2 ) ]−5 [ (7∗6 )−(3∗1 ) ]+4 [ (7∗2 )−(0∗1 ) ]

−1 (6 )−5 (45 )+4 (14 )=−6−225+56=−175

3.- [ 2013005−1−1]

2 [ (0∗1 )−(−1∗0 ) ] 0 [0 ]+1[ (3∗−2 )−(5∗0)]

2 (0 )0 (0 )+1 (0 )=0

4.- [2−10710131111 ]

2 [013277111 ]+ 1 [113377

111 ]+0 -7 [101327111]

0 [0 (7∗7 )−1 (2∗−7 ) ]+3[ (2∗−7 )] = -20

1 [1 (7∗−7 )−1 (3∗−7 ) ]+3 [ (3∗−7 )] = -8

−7 [1 (2∗−7 )+1 (3∗−2 ) ] = -28

R=-20-8-28= - 56

5.- [ 1−211302−204−1−1503−1 ]

1 [ 02−24−1−103−1 ] +2 [ 32−20−1−1

53−1 ]+1 [30−204−150−1] - 1 [30−204−1

50−1]

1 [0 (4 )−2 (−4 ) ]−2 (12 )¿ = -16

2 [3 (4 )−2 (5 ) ]−2(5)¿ = -16

1 [3 (−4 )0 (−0 ) ]−2(−20)¿ = 28

−1 [3 (12 ) ]−2[ (−20 )] = -4

R=-16-16+28-4=0

6.-[ 1−211302−204−1−1−16−20]

1 [ 02−24−1−10−20 ] +2 [ 32−20−1−1

−1−20]+1 [30−204−150−1] + 1 [30−204−1

−160]-1 [ 30−20 4−1−16−2]

1 [0 (−2 )−2 (0 ) ]−2 (8 )¿ = -16

2 [3 (−2 )−2 (0 ) ]−2(−1)¿ = -8

1 [3 (6 )0 (0 ) ]−2(4)¿ = -4

−1 [3 (−6 ) ] 2[(4 )] = -2

R=-8-16-8-4-2=-38

7 20 23 14 1

17 13 4 13 4 13 4 15 4 5 2 2

26 11 5 2 2 7 11 5 2 2 13 22 14 25

30 26 22 14 25 26 22 14 25 19 22 8

5 8 19 22 8 8 19 22 8

5(112-550)-2(176-475)+2(484-256)= -438-299 218=1156*13 -2190+598 456=15028

7.-

11 2 2 11(112-550)-2(208-200)+2(572-112) = 4818-16+920-3914*-4=15,256

-4 26 14 26

8 22 8

11 5 5 11(176-475)-5(208-200)+2(494-176)= -3289-40+636=2693*13=-35009

13 26 22 25

8 19 8

11 5 2 11(484-266)-5(572-112)+2(497-176)2398-2320+636=714*4=-2856

-4 26 22 14 =-37237*2=-260,654

8 19 20

17 4 13 4

20 26 5 2 2 5 2 2

30 22 14 25 17 22 14 25

5 19 22 3 19 22 8

5(112-550)-2(176-475)+2(484-266)-2190+598+436= 1156*17=-19652

26 2 2

-4 30 14 25 26(112-550)-2(246-125)+2(660-70)-11388-230+1180=10438*-4=41752

5 22 8

13 26 5 2

30 22 25 26(176-475)-5(240-125)+2(570-110)-7774-575-920=-10438*13-96577

5 19 8

-4 26 5 2

30 22 14 26(484-256)-5(560-70)12(570-110)-3638+5668-2950+420=70839*20

6 14 22 114261780

17 13 13 4 11 2 2

23 26 11 2 2 17 20 14 25

30 26 14 25 5 22 8

5 8 22 8

11(112-550)-2(208-200)+2(572-112)-4818-16+920

=-391*17=-66538

26 2 2

-13 30 14 25 26(208-200)-11(240-125)+2(240-130)=208-1265+220=-837*13=-10881

5 22 8

26 11 2

13 30 26 25 26(208-200-11(240-125)+2(240-130)=-837*13=-10881

5 8 0

26 11 2

-4 30 26 14 26(572-112)-11(660-70)+2(240-430)=5696*-4=-22760

5 8 22 35515*23=8161845

17 13 4 4 11 5 2

-14 26 11 5 2 17 26 22 25 11(176-475)-5(208-200)+2(494-176)-3289-

30 26 22 25 8 19 8 -40+636=-2693*17=-45781

5 8 14 8

26 5 12

-13 30 22 25 -26(176-475)-5(240-125)+2(570-110)=7429*13-7774-575+920

5 19 8 =96577

26 11 2

4 30 26 25 26(208-200)-11(240-125)+5(240-130)-208-1265+220=837*4=-3348

5 8 8

26 11 5

-4 30 26 22 26(494-176)-11(570-110)+5(420-130)8268-5060+550=

5 8 19 3758*-4=15032 =32410*-14=453824

17 13 4 13 11 5 2

26 11 5 2 17 26 22 14 11(484-266)-5(572-112)+2(494-176)

30 26 22 14 8 14 22 2398-2300+636=734*17=12478

5 8 14 22

26 5 2

-13 30 22 14 26(482-266)-5(660-70)+2(570-110)5668-2950+920

5 19 22 =3638*13=47294

26 11 2

4 30 26 14 26(572-112)-11(660-70)+2(290-130)11960-6490+220=5690*4

5 8 22 =22760

26 11 5 26(494-176)-11(570-110)+5(240-130)8268-5060+550

-13 30 26 22

5 8 19 =3758*13 -48854-60410= 1,821472

2 3 4 1 = 8 20

-1 2 0 6 -4 11

(2) (4) + (3) (0)= 28

(2) (1) + (3) (6)= 20

(-1) (4) + (2) (0)= -4

(-1) (1) + (2) (6)= 11

3 -2 -5 6 = -17 12

1 4 1 3 -1 18

(3) (-5) + (-2) (1)= -17

(3) (6) + (-2) (3)= 12

(1) (-5) + (4) (1)= -1

(1) (6) + (4) (3)= 18

1 -1 1 0 = -1 -3

1 1 2 3 1 3

(1) (1) + (-1) (2)= -1(1) (0) + (-1) (3)= -3

8.-

9.-

10.-

(1) (-1) + (1) (2)= 1(1) (-1) + (1) (3)= 3

-5 6 3 -2 = -9 14

1 3 1 4 6 10

(-5) (3) + (6) (1)= -9

(-5) (2) + (6) 4)= 14

(1) (3) + (3) (1)= 6

(1) (-2) + (3) (4)= 10

3 -1 1

-4 5 1 5 6 4 = 13 35 18

0 4 2 0 1 2 20 26 20

(-4) (3) + (5) (5) + (1) (0)= 13

(-4) (-1) + (5) (6) + (1) (1)= 35

(-4) (1) + (5) (4) + (1) (1)= 18

(0) (3) + (4) (5) + (2) (0)= 20

(0) (-1) + (4) (6) + (2) (1)= 26

(0) (1) + (4) (4) + (2) (2)= 20

1 6 7 1 4 = 19 17 34

11.-

12.-

13.-

0 4 2 -3 5 8 -12 20 (-2) (7) + (-2) (2)= -18

-2 3 -18 4 -18 (-2) (1) + (-2) (-3)= 4

(-2) (4) + (-2) (5)= -18

(1) (7) + (6) (2) = 19(1) (1) + (6) (-3)= -17(1) (4) + (6) (5)= 34(0) (7) + (4) (2)= 8(0) (1) + (4) (-3)= -12(0) (4) + (4) (5)= 20

[1 −11 1 ] [ 0 2

−1 1]=[ 1 1−1 3]

0 [11]+ (−1 )[−11 ]=0+[ 1−1]=[ 1−1]2[11]+1[−11 ]=[22]+[−11 ]=[13]

[ 4 1−2 3] [−1 0

2 3]=[−2 38 9]

−1[ 4−2]+2 [13]=[−42 ]+[26]=[−28 ]0 [ 4−2]+3[13]=[00]+[39]=[39 ]

[5 −4 02 4 1] [1 0 1

6 4 25 −1 3 ]=[−19 −16 −3

31 15 13 ]1[52]+6 [−44 ]+5[01]=[−1931 ]0 [52]+4 [−44 ]+ (−1 )[01]=[−1615 ]1[52]+2[−44 ]+3[01]=[−313 ]

14.-

15.-

16.-

[ 2 −1 5−4 0 4 ][ 3 2 1

5 0 1−1 6 4]=[ −4 34 21

−16 16 12]3[ 2−4]+5 [−10 ]+(−1 )[54 ]=[ −4

−16]2[ 2−4]+0 [−10 ]+6 [54]=[3416 ]1[ 2−4 ]+1[−10 ]+4 [54]=[2112]

[4 63 01 −2] [2 4 5

3 1 −1]=[ 26 22 146 12 15

−4 2 7 ]2[431 ]+3 [

60

−2]=[ 266−4 ]4 [431 ]+1[

60

−2]=[22122 ]5[ 431]+ (−1 )[ 60−2]=[14157 ][2 −13 15 4 ] [1 0 3

4 6 −2]=[−2 −6 87 6 721 24 7 ]

1[235]+4 [−114 ]=[−2721 ]

0 [235 ]+6[−114 ]=[−6624 ]

17.-

18.-

19.-

3[235]+ (−2 )[−114 ]=[877]

[4 1 −20 3 54 6 1 ][1 6 2

3 0 15 −3 2]=[ 6 10 13

28 5 1942 13 4 ]

1[404 ]+6 [136]+2[

−251 ]=[ 62842]

3[ 404]+0[136]+1[

−251 ]=[10513]

5[ 404]+ (−3 )[136]+2[−251 ]=[13194 ]

[0 −1 11 3 24 −2 0][

0 3 12 4 01 −1 −2]=[−2 −4 −1

11 14 −6−6 0 6 ]

0 [014]+3[−13

−2]+1[120]=[−211−6 ]

2[014 ]+4[−13

−2]+0[120]=[−4140 ]

1[014 ]+(−1 )[−13−2]+ (−2 )[120]=[−1−66 ]

[4 6 13 0 −11 −2 4

235

2 30 −2 ][ 3 4 15 6 0

−2 1 −1

−212

2 04 3 ]=[ 33 57 12 14 1

−3 −3 2

181133

22 28−5 −2]

20.-

21.-

22.-

3[ 4312]+4 [60

−23 ]+1[ 1−140 ]+ (−2 )[ 235

−2]=[ 332−322 ]5[ 4312]+6 [

60

−23 ]+0[ 1−140 ]+1 [2352]=[5714−3

28 ]−2[4312 ]+1[

60

−23 ]+(−1 )[ 1−140 ]+2[2352]=[ 112

−5]2[4312 ]+0 [

60

−23 ]+4 [ 1−140 ]+3[2352]=[ 181133

−2]33[ 14 1 11

−3 2 3328 −5 −2]=¿

¿33 [14 [ (2∗(−2))−(33∗(−5)) ]−1 [ ( (−3 )∗(−2 ) )−(33∗28 ) ]+11 [ ( (−3 )∗(−5 ) )−(2∗28 ) ] ]=¿

¿33 [2254−918+451 ]=33 [1787 ]=58971

57 [ 2 1 11−3 2 3322 −5 −2]=¿

57 [2 [ (2∗(−2 ) )−(33∗5 ) ]−1 [ ( (−3 )∗(−2 ) )−(33∗22 ) ]+11 [ ( (−3 )∗(−5 ) )−(22∗2 ) ] ]¿57 [338−720+319 ]=57 [−63 ]=−3591

1[ 2 14 11−3 −3 3322 28 −2]=¿

1 [2 [ ( (−3 )∗(−2 ) )−(33∗28 )−14 ( (−3 )∗(−2 ) )− (33∗22 ) ]+11 [ ( (−3 )∗28 )−( (−3 )∗22 ) ] ]

23.-

24.-

25.-

¿ [1836−10080+198 ]=−8046

18[ 2 14 1−3 −3 222 28 −5]=¿

18 [2 [ ( (−3 )∗(−5 ) )−(2∗28 ) ]−14 [ ( (−3 )∗(−5 ) )−(2∗22 ) ]+1 [ ( (−3 )∗28 )−( (−3 )∗28 )−( (−3 )∗22 ) ] ]¿8 [82−406+18 ]=−2448

58971-3591-8046-2448= 49782

METODO DE JORDAN

⌊1 1 −14 −1 52 2 −3

740⌋

⌊1 1 −10 −5 90 0 −1

1−24−14

⌋

26.-

2-. X1 + X2 – X3 =7

4X1 – X2 + 5x3 =4

2X1 + 2X2 -3X3 =0

R2_R2_-4 R1

R3_R3_-2 R2

R2- -1/5R2

⌊1 1 −10 1 −9 /50 0 −1

1−24/5−14

⌋

⌊1 1 −10 1 −9 /50 0 −1

11 /5−24/5−14

⌋

⌊1 1 00 1 00 0 1

−93014

⌋

[−2 1 63 2 −105 0 8

¿18¿−3¿−16 ]R1→−1

2R1[1 −1

2−3

3 2 −105 0 8

¿−9¿−3¿−16]R2→R2−3R1

R2- -1/5R2

R1-R1 -4/5R2-R2 9/5

¿

¿0 72−124

−2 x1+x2+6 x3=18

3 x1+2 x2−10 x3=−3

5 x1+8 x3=−16

[1−12

−3

0 72

−1

5 0 8

¿−9¿24

¿−16]R3→R3−5 R1[1−12

−3

0 72

−1

0 52

23

¿−9¿24¿29 ]R2→ 2

7R2[1

−12

−3

0 1 −27

0 52

23

¿−9

¿ 487

¿29 ]R3→R3−52R2

[1−12

−3

0 1 −27

0 0 1667

¿−9

¿ 487

¿ 837 ]R3→ 7

166R3[1

−12

−3

0 1 −27

0 0 1

¿−9

¿ 487

¿ 12

]R2→R2+27R3[1 −1

2−3

0 1 00 0 1

¿−9¿7

¿ 12 ]R1→R1+3 R3

[1 −12

0

0 1 00 0 1

¿−132

¿7

¿ 12

]R1→R1+12R2[1 0 00 1 00 0 1

¿−82

¿7

¿ 12

]=−82

=X17=X212=X3

001 12¿

010 4971−120−13

2

¿

¿100−82

[1 −2 34 1 −12 −1 3

¿11¿4¿10]R2→R2−4 R1R3→R3−2 R1 [1 −2 3

0 9 −130 3 −3

¿11¿−40¿−12]R2→ 1

9R2

[1 −2 3

0 1 −139

0 3 −3

¿11

¿−409

¿−12 ]R3→R3−3 R2[1 −2 3

0 1 −139

0 0 43

¿11

¿−409

¿ 43

]R2→ 34R3[1 −2 3

0 1 −139

0 0 1

¿11

¿−409

¿1 ]R2→R2+ 139 R3

[1 −2 30 1 00 0 1

¿11¿−3¿1 ]R1→R1−3 R3[1 −2 0

0 1 00 0 1

¿8¿−3¿1 ]R1→R1+ 12 R2[1 0 0

0 1 00 0 1

¿2¿3¿1]

x1−2 x2+3 x3=11

4 x1+x2−x3=4

2 x1−X2+3 x3=10¿

¿09−13−4003−3−12

¿

00 434300 166

7837

¿

0−2081002

x1−2x 2+6 x3=10

5 x1+8 x3=3

3 x1−10x 2+2x 3=−3

1 −2 6 105 0 8 33 −10 2 −3

1 −2 6 103 −10 2 −35 0 8 3

[1 −2 6 103 −10 −10 −35 0 8 3 ]3R1[1 −2 6 10

0 −4 −16 −335 0 8 3 ]5R1[1 −2 6 10

0 −4 −16 −330 10 −22 −47 ]R2/4

[1 −2 6 10

0 1 4 81/4

0 10 −22 −47]−10 R2[1 −2 6 10

0 1 4

814

0 0 −2 −147]+2R2[1 0 14

2612

0 1 4

814

0 0 −2 −147]R3/−2[1 0 14

2612

0 1 4

814

0 0 1

7314

]−4 R3 [1 0 14 261

/2

0 1 4 −2853

/4

0 0 1 731

/2 ]−14 R3[1 0 0 −10021

/2

0 1 0 −2853

/4

0 0 1 731

/2 ]

2

3

X1=-−10021

/2

X2=−2853

/4

X3=731

/2

(−10021

/2 )-2(−2853

/4)+6(731

/2)= 10

5(−10021

/2 ) +8(731

/2)= 3

3(−10021

/2 )-10(−2853

/4)+2(731

/2)= -3

⌊4 3 21 3 −5

−3 −3 8⋮123⌋R1→ 1

4R1=⌊

1 34

24

1 3 −5−3 −3 8

⋮

1423

⌋ R2→R2−1 R1 ⌊1 3

424

0 94

−224

−3 −3 8

⋮

14743

⌋

R3 3/23 R3

4 X1+3 X2+2 X3

X1+3 X2−5 X3

3 X1−3 X2+8 X3

R3 R3+3R1

R11 342414

R21 3 −5 2

−1 −34

−24

−14

0 94

−224

74

R3−3 −3 8 3

3 94

64

34

0 −34

384

154

4(-152/92)+3(149/69)+2(13/23)=1

1(-152/92)+3(149/69)-5(13/23)=2

-3(-152/92)-3(149/69)+8(13/23)=3

⌊

1 34

24

0 94

−224

0 −34

384

⋮

1474154

⌋ R2→ 49R2 ⌊

1 34

24

0 1 −229

0 −34

384

⋮

1479154

⌋ R3→R3+ 34R2 ⌊

1 34

24

0 1 −229

0 0 233

⋮

1479133

⌋

⌊1 34

24

0 1 −229

0 0 1

⋮

14791323

⌋R2→R2+ 229⌊1 3

424

0 1 00 0 1

⋮

14149691323

⌋R1+(−24 R3) ⌊1 3

40

0 1 00 0 1

⋮

−392149691323

⌋

⌊1 0 00 1 00 0 1

⋮

−15292149691323

⌋

X1= -152/92

X2=149/69

X3=13/23

R1 R1-3/4R2

R201−22979

R30 −34

384

154

0 34

−116

712

0 0 233

133

R3001 1323

R20 1 −229

79

0 0 229

286207

0 1 0 14969

R11 34

24

14

0 0 −24

−1346

1 34

0 −392

R11 340 −3

92

0 −34

0 −14992

10 0 −15292