Fasciculo6

1

!

"

#

$

!

% &

' ()

!

% + ,

-

$

*(

*

.

&

/ 0 / /1 /#

0 / 0 /

/ # $

0 /

/ #

0 /

/ 2

!

/ 2

! −

/

3 2

2 0 / 3 −3

%/ 3 1

0 /

& / −

3 3 − = 2 2

/ −

3 3 + = 2 2

/ −

1 (− 2) = 2

/ 7

5 3 − 7 2

−

1 0 / 3

4 = 9

/ (−5)1 + 5 = / (−3)

5 1 − (−3) + (−3)(4) = 9 2

Se pueden efectuar

como

para solucionar

4 15 65 2 ,728 9 : 5 86

2

que puede ser

0 /

%

"

!

;

" ; "

$

5

" "

<

"

M=

< m× n =

m11

m12 . .

.

m1n

m21

m22

.

.

m2 n .

.

.

.

.

.

.

.

. .mij

mm1

mm 2

m

.

.

#

m ij

M m ×n

. mmn

i − esima j − esima

m ×n

n

3

−1 − 2 2 1 R= 3 −3 4 −1

3 4 0 −3

> %3 4×3

2 1 4

R32

(−3)

R42

H=

4

2 0 80 − 30 40 − 50 − 10 H

<

2× 3

H

+ %3

+

3

&

a12 = 80 a23 = −10

P = [ Pij ]

Q = [Qij ]

Py Q

j

i Pij = Qij

2 −5 − 27 0

4

2× 2

1 3 + 2 2 = (− 3)3

7 − −

5 7

3 3 + 4 4

2×2

;

#

5 + 3z − 6z − 2

2 +z 3 4z

6

=

−4

2×2

1 4 3

&

2 + z =1 3 4 4z = 3

5 + 3z = 6 − 6 z − 2 = −4

2×2

$

1 &

5 + 3z = 6

2 + z =1 3

3z = 1 z=

1 3

z = 1− z=

− 6 z − 2 = −4 2 3

− 6 z = −2

1 3

z=

1 3

4 3 4 z= 12 1 z= 3

4z =

x, y, w, z

,

5x + 4 2 7 + 2x 6w = y − 7 3z 12 15 − 2 z &

5x + 4 = 7 + 2 x 5x − 2 x = 7 − 4 x =1

2 = 6w 2 =w 6 1 =w 3

y − 7 = 12

3z = 15 − 2 z

y = 12 + 7 3z + 2 z = 15 y = 19

z=3 &

1 x = 1, w = , y = 19, z = 3 3

1 3 + 2 2 −5+ w 3 z 4

2 − 3x − C = 7 y +1

(− 2)3

8 !

H=

1

6 −4 −8 −3

5

" #

$%&

&(

C32

&)

H 2×3 $ & 3 (− 2) $ &

5 7 & H 31 = C31 $ &

&

$'&

C 21 = H 21 $ &

y=

H =C

x, y, w, z

* + , -

!

. -

!

/ -

!

4×3 !

2×3

-01'1# #

(02

(

!

3

!

4 !

4

4

4 4 3

3 !

6 3

3 ( 5

! ! ) ,. 7-

4

8

3 !

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2

6

&

!

n

&

"

"

! n

n×n

M=

m11

m12 . .

.

m1n

m21

m22

.

.

m2 n .

.

.

.

.

.

.

.

. .mij

mn1

mn 2

.

m11 , m22 , m33 , !

.

n×n

. mnn

n×n

mnn

=

mij

i= j

&

i≠ j !

mij

m×n

&

m×n !

n×n

&;

mij

0'/

i≠ j i= j

n !

P T = [ Pij ]

m×n

P = [ Pij ]

&

n×m

i − ésima

i − ésima

P

2

7

P

8

F = 1 −2 9 3 5 4

G= 3×3

PT

5 0 0

20 19 70 12 13 80

2×3

H= 0 3 0 0 0 7

3×3

7

I4 =

1 0 0 0 0 1 0 0

J= 0 0 0 0

0 0 1 0 0 0 0 1

F

0 0

4×4

3×2

3× 3 H

?

3× 3 4× 4 J

I4

3× 2

20 12 G = 19 13 70 80 T

G 3×2

3 !

3 4

!

u = [1 , − 3]1×2 v = [7 , 5]1×2 p=

2 5

q= 2×1

4 3

2×1

r = [2, 5,−7] s = [7, − 2, 8] t = [9, 6, 5]

6 !

u + v = [1 + 7,−3 + 5] = [8 , 2]

p+q =

2+4 6 = 5+3 8

r + s + t = [2 + 7 + 9, 5 − 2 + 6,−7 + 8 + 5] = [18, 9, 6]

8

= #

!

M , N, S

M=

m11

m12 . .

.

m1n

n11

n12 . .

.

n1n

m21

m22

.

.

m2 n .

n21

n22

.

.

n2 n .

.

.

.

.

.

.

.

.

.

.

.

.

. .mij

.

.

. .nij

.

mm1

mm 2

nm1

nm 2

.

N=

. mmn

.

m ×n

.

nmn

.

m11 + n11

m12 + n12

.

.

m1n + n1n

m21 + n21

m22 + n22

.

.

m2 n + n 2 n .

.

.

.

.

.

.

.

. .mij + nij

mm1 + nm1

mm 2 + nm 2

M +N =S =

.

. mmn + nmn

.

m ×n

m ×n

3 -

3

+ 9

v = [−3, 5, 8] !

(− 2.5) (− 2.5v ) = [7.5, − 12.5,−20] "

! k

kM M

k

m×n

M = [mij ]m×n

"

kM

m×n

"

"

M

k

9

#

$

H=

&

(−3) H = (− 3)

−2 4 9 −1 3 5

2 1 4 3

−2 4 9 6 − 12 − 27 = −1 3 5 3 − 9 − 15

3 3 3 −2 4 9 − H= − = 3 2 2 −1 3 5 2 5I = 5

I=

−6 −

9 2

27 2 15 − 2

−

2 1 10 5 = 4 3 20 15

#

:

u = [u1 , u 2 , u3 ]1×3 ! v = [v1 , v2 , v3 ]1×3 u•v k u

v

u • v = [u1 , u 2 , u3 ] • [v1 , v2 , v3 ] = u1 ⋅ v1 + u 2 ⋅ v2 + u3 ⋅ v3 = k 6

2

−1

5 3

• −2 −3 3×1

5

[3, 5,−4] • [−2,7,−1] = −6 + 35 + 4 = 33

= −2 − 10 − 9 = −21 3×1

[−2,11]1×2 •

#

−3 −5

2×1

" −2 11

10

• 2×1

−3 −5

= 6 − 55 = −49 2×1

" m×n

P = [ Pij ] P ⋅Q = R

n×r

Q = [Qij ]

n

"

n

"

P

m×r

R = [ Rij ]

Q

P ⋅Q

" "

P

"

Q

−4 #

$

u = [−1, 7, 3] v = [−3, 6,−1] w = − 3

&

−5 L=

N=

2

−1 4 −3 5

2× 2

−1 6

M = 8 −3 4 5 −5 7

−3 2 −4 −1 7 − 5

2×3

3×3

2 −5

1 − 3 11 1

Ñ = 1 −3 0 −2

O = 2 − 2 13 8 3 − 4 10 7 3×2

3×4

u • v = (− 1)(− 3) + 7 ⋅ 6 + 3(− 1) = 3 + 42 − 3 = 42 L⋅N =

4

−1 4 −3 5

−3 2 −4 2×2

=

−1 7 − 5

2×3

− 1 26 − 16 4

29 − 13

2×3

(− 1)(− 3) + 4(− 1) = −1 @ (− 1)2 + 4 ⋅ 7 = 26 @ (− 1)(− 4) + 4(− 5) = −16 (− 3)(− 3) + 5(− 1) = 4 N ⋅L O⋅M

@ (− 3)2 + 5 ⋅ 7 = 29 @ (− 3)(− 4) + 5(− 5) = −13 N O

3

>

L

+

M

3

11

4

&

−1 6

1 − 3 11 1

18 − 28 69 36

M ⋅O = 8 − 3 4 5 −5 7

2 − 2 13 8 3 − 4 10 7

= 14 − 34 89 12 16 − 33 60 14

2

3×3

2 ⋅1 + (− 1)2 + 6 ⋅ 3 = 18

@

2 ⋅11 + (− 1)13 + 6 ⋅10 = 69

@

8 ⋅1 + (− 3)2 + 4.3 = 14

@

8 ⋅11 + (− 3)13 + 4 ⋅10 = 89

@

5 ⋅1 + (− 5)2 + 7 ⋅ 3 = 16

@

5 ⋅11 + (− 5)13 + 7 ⋅10 = 60

@

L⋅M

3×4

2(− 3) + (− 1)(− 2) + 6(− 4) = −28 @ 2 ⋅1 + (− 1)8 + 6 ⋅ 7 = 36 8(− 3) + (− 3)(− 2) + 4(− 4) = −34 8 ⋅1 + (− 3)8 + 4 ⋅ 7 = 12 5(− 3) + (− 5)(− 2) + 7(− 4) = −33 5 ⋅1 + (− 5)8 + 7 ⋅ 7 = 14

M ⋅L

3LN − 2 Ñ T = 3

:

3×4

−1 4 − 3 2 − 4 2 1 0 = −2 −5 −3 −2 − 3 5 −1 7 − 5

− 3 12 − 3 2 − 4 4 2 0 − 3 78 − 48 4 2 0 − = − = − 9 15 − 1 7 − 5 − 10 − 6 − 4 12 87 − 39 − 10 − 6 − 4 − 7 76 − 48 22 93 − 35

#

−1

7

5

d = [1, 3,−5] e = [4,−7, 3] f = − 2 K = 2 − 1

−7 5 −7 2 −2 1 G= H = 7 −4 3 −3 5 8 −1 5

12

& − 3e

" 2

& 5d

* 2

& 3 I T + 2k

, 2

& I ⋅ J & 7J ⋅ I

&

&d ⋅ K

2 6 −1 I= 8 7 −2 1 f 2

& 7 d − 4e & & I⋅f

&K ⋅H

1

4

−1 0 7 3 J = −3 2 4 8 −2 4 0 4 3 K 2

&H ⋅J

& 3G ⋅ I − 2 K T

$ !

5 > <

2× 2

3× 3

<

;

! <

!

9

A

!

,

&

a11 x1 + a12 x2 + ....... + a1n xn = b1 a21 x1 + a22 x2 + ....... + a2 n xn = b2 . . . am1 x1 + am 2 x2 + ....... + amn xn = bm <

m×n

2

2B

a11 a12 ... a1n a21 a22 ... a2 n . . . . . . . . . am1 am 2 ... amn &

13

x1 x2 .

X = . . xn

b1 b2 .

B= . . bm

2 ) a11

a12

... a1n b1

a21 .

a22 .

. .

. .

... a2 n b2 . . . . . .

am1 am 2 ... amn bm

2 % &

1

'

"

a 1 a

1 a =1 a

+

a11

3 ? "

' !

0

/

!

0

1

"

a

(− a ) a + (− a ) = 0

&

/ ;

14

a22 0

>

C

/

"

!

'

D

1

!

09

A

/

w + 3m + 2n = 3 3w − 2 m − n = 1 )

4 w − m + 3n = −4

a21 y a31 1

&

a11

'0

3

2 3

3 − 2 −1 1 4 −1 3 − 4

/

0 F2 → −3F1 + F2 / 0 F3 → −4 F1 + F3 / 1

3

0 − 11 0 − 13

2

3

−7 −8 − 5 − 16

& 3 E 30'/ B F @ E+ E 303/ B E ''@ E ' E 30+/

B E G @ 'E 303/ B E H > E >0'/ B F @ E ' E >03/ B E '3@ 3E >0+/ B E C @ E >E >03/ B E 'D E ''

'0

E ''

+/

15

1

3

2

3

0 F1 → −3F2 + F1 /

7 8 0 1 11 11 0 − 13 − 5 − 16

1 0

9 11 8 11 72 11

1 11 7 11 36 11

−

3 − 3(118 ) =

9 11

0 1 0 0

− 16 + (13)(118 ) =

& ' E 30F/ B '@ 3 E 30'/ B F @ 2 − (3)(117 ) =

1 11

@

F I '3 0F/ B F @ E '3 I '30'/ B F @ − 5 + (13)(117 ) =

36 11

@

−72 11

3DJ''

1 0 0 1 0 0

1 11 7 11

'0

9 11 8 11

0 F1 →

−1 11

''J3D

F3 + F1 /

0 F2 →

1 11

−1 + (11 )(1) = 0 @

−1 + (11 )(− 2) = 1

& 0 + ( −117 )(0) = 0 @ 1 + ( −117 )(0) = 1 @

7 11

& K B '@

1

!

+ ( −117 )(1) = 0 @ 118 + ( −117 )(− 2) = 2 B +@

09

B E+

A

/

5 − 11 43 4 22

5m − 11n = 43 4m + 2n = 2 5'

?

16

F3 + F2 /

−1 & 1 + ( 11 )(0) = 1 @ 0 + (11−1 )(0) = 0 @

0 1 0 2 0 0 1 −2

:

−7 11

3/

1 −2

1 0 0 1

9 11

0 F3 → 13F2 + F3 /

1

−11 43 5 5

4

22

L

F2 → −4 F1 + F2

1 5

1

−11 5

0

54 5

43 5

−

1(− 4) + 4 = 0 @ (− 4)( −511 ) + 105 =

162 5

@ (− 4) 435 + 105 = − 162 5

5 54

?

1 0 11 5

54 5

−11 5

43 5

F1 → 115 F2 + F1

1 −3

⋅ 0 +1 = 1 @

11 5

⋅1 − 115 = 0 @

11 5

1 0 2 0 1−3

(− 3) + 435 = 2 m=2

y n = −3

9 ;

<

:

(

"

10 x + 10 y = 100

&

10 x − 10 y = 80 30 x + 30 y = 50

&

20 x + 20 y = 100

2 x − 6 y = 40

&

4 x + 2 y = 50

* ;

<

:

2x − 3 y − 2z = 2 & x − y − 5 z = 18

x+ y−z = 6 m − n − 3z = −4 & 2 m − n − 4 z = −7

m + n − z = −2

17

3

=

#

!

!

−2 u = [−2,−5] v = [3,−5,−1] w = − 6

4 −1 1 C= 0 2

A=

1 −2 0 3

B=

−2 3 0 1 −4 1

1 0 0

3 D= 0 1 1 4 1 2 1

" -

9 & − 3u

& 3v

& 5v • w

&u ⋅ A &v⋅ D

* 2

: & 3 A − 2 BC & BT B & B (D + E ) & D 2

,

7 m + 2n = 8 3m − 5n = 21 .

;

2w + 3z = 17 3w + 4 z = 24

" &

18

><

:

&

& <

;

&

?

"

& "

#

&

#

& "

"

. "

!

9

4

A

. ?

2

-

#

) 4

:

+FF'

?

L

=4 : ? # : 5 'MMM ) L

,

N

O

6

2

P

'MM+

19

,4

9

O

)

) .

? 9 1

5 <

?

!

): O

2 K4 2

+FF' .

+FF+

) P

.

Q

O

Q?

2 ?

9

K 4

=

#

$ $

20

'MMH

$

5

> 5 C ,

) 5

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2

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& RRRRRRRRRRRRRRRRR & RRRRRRRRRRRRRRR

' :

&

/; / /, /, /,

0 / 0 / 0 / 0 / 0 /

,

&

1 u = [−7,−2] v = [9,−5, 6] w = [−8,7,−1] z = 3

5 −3 −2 A= 0 −5

−1 1 1 2 0 −2 0 3 B= C= 0 3 D= 0 1 1 −1 − 4 1 2 4 1 0 1

3 0 0

1 3

0 0

1 0 0

E = 0 6 0 F = 0 16 0 I 3 = 0 1 0 0 0 3 0 0 13 0 0 1 v•w

+ 2 H

O

2H

O 'M

v• z

3

>: C 4

&

''3

)

'FC

, '+'

& ) ''3

, +>

& 4D − 2E & / 3 A − 2 BC

/ DI − 13 E / D 2

/ B (D + E )

/ EF

21

"

*

* 2 O ) ,

D #

5 x − 6 y = 13 7 x + 3 y = −16

9

'

+

A

3

2

3

O

x

)

,

y

G #

12m + 3n = 33 3m + 14n = 48

9

'

+

2

3 O

m

A

3 )

,

n

H ) !

!

9

M 1

A

!

9

A

2 x − 3 y + z = −2 3x + 3 y − 2 z = 2 x − 6 y + 3 z = −2 'F ) !

22

!

9

A

&