ACTIVIDAD DE CLASIFICACION DE LOS NUMEROS

...

1 0 3

[pic 31][pic 32]

1 0 3 x=3

0 1 1 y=1

OPERACIONES CON MATRICES: SUMA ,RESTA,MULTIPLICACION X ESCALAR

SUMA Y RESTA TAMAÑO R X K

[pic 33][pic 34][pic 35]

A 1 -2 3 A+B 3+1 -2+0 3+2 = 4 -2 5

4 5 -6 4-7 5+1 -6+8 -3 6 2

[pic 36]

B 3 0 2

-7 1 8

[pic 37][pic 38][pic 39][pic 40]

3A 3 -6 9 2A – 3B 2 -4 6 - 9 0 6 = -7 -4 0

- 15 -18 8 10 -12 -21 3 24 29 7 -36

MULTIPLICACION: NUMERO DE COLUMNASQUE COINCIDEN CON EL NUMERO DE FILAS SI SE PUEDE MULTIPLICAR.2 X 2 2X2 A(MXP) B (PXN)=MXN[pic 41][pic 42]

1 2 5 0 = 1X5+ 2X6 1X0+2X7 = 5+(-12) 0+12 = -7 14

3 -4 -6 7 = 3X5+-4X-6 3X0+-4X7 = 15+24 0+(-28) = 39 -32

DETERMINANTES [pic 43]

A= a11 a12 = a11 a22-a12 a21[pic 44]

a21 a22

[pic 45][pic 46]

5 4 = 5(3)-4(2)= 15-8=7

1 3

[pic 47]

2 1 =2(6)-(-4)(1)=12+4=16[pic 48]

-4 6

a 13 a 11 a 12 a 13 a11[pic 49][pic 50]

a 21 a 22 a23 = a 13,a 21,a 32+a 11, a 22, a33+a12,a23,a31-a12,a12,a33-

a 33 a31 a32 a33 a31 -a13,a22,a31-a11,a23,a32

[pic 51][pic 52]

3 1 2 3 1

4 -2 3 = 3(4)(5)+1(-2)(-1)+2(3)-2(4)(-1)-3(-2)(0)-1(3)(5)=60+2+6-15=55

-1 0 5 -1 0

REGLA DE CRAMER : 1.- CALCULAR (D) DE LA MATRIZ DE COEFICIENTES

2.- SUSTITUIR LOS TERMINOS INDEPENDIENTES EN X,Y,Z Calculando (NX),(NY),(NZ)

3.-SOLUCION X=(NX)/(D) Y= (NY)/(D) Z=(NZ)/(D)

(D)= DETERMINANTE DE LA MATRIZ

X-2Y+Z=7 2X-Y+4Z=17 3X-2Y+2Z=14[pic 53]

1 1 -2 1 1[pic 54]

2 -1 4 =1(2)(-2)+7(-1)(2)+(-2)(4)(3)-(-2)(2)(2)+1(-1)(3)+1(4)(-2)=-4-2-24+8+3+8=30+19

2 3 -2 2 3 =-11

[pic 55]

(NX) 1 7 -2 1 7[pic 56]

17 -1 4 =17(1)(-2)+7(-1)(2)+(-2)(4)(14)-(-2)(17)(2)-(-1)(-1)(14)-7(4)(-2)=

2 14 -2 2 14 =-34-14-112+68+14+56=-22

[pic 57][pic 58]

(NY) 1 1 7 1 1

2 17 4 =1(2)(14)+1(17)(2)+7(4)(3)-7(2)(2)-(-1)(17)(3)-1(4)(14)=

2 3 14 2 3 =28+34+84-28-51-56=11

(NZ) 1 1 -2 7 1[pic 59][pic 60]

2 -1 17 =-28-14-102+56+21+34=-33

14 3 -2 14 3

X=-22/-11=2 Y=11/-11=-1 Z=-33/-11=3

TRASPUESTA E INVERSA DE UNA MATRIZ

TRASPUESTA= ES UNA MATRIZ RESULTANTE DE INTERCAMBIAR RENGLONES POR COLUMNAS

A= 1 -3 AT= 1 6[pic 61][pic 62]

6 5 -3 5

[pic 63]

A= 1 2 3 AT= 1 4 7[pic 64]

4 5 6 2 5 8

7 8 9 3 6 9

A= 2 0 -3 AT= 2 5[pic 65][pic 66]

5 -1 4 0 -1

-3 -4

A a1= matriz identidad

[pic 67]

1 0 1 0 0[pic 68]

0 1 0 1 0

0 0 1

POR ADJUNTOS Y DETERMINANTES

A1=(AD)T/(A)

[pic 69]

A= 1 3 2 A=2 1 -3 2 1[pic 70]

2 5 0 2 5 0 =-4-10-12=-26

0 -1 -2 -2 0 -1 -2 0

[pic 71]

(AD) A11= + 5 0 = -10 A12= - 2 0 =4 A13=+ 2 5 =-2[pic 72][pic 73]

-1 -2 0 -2 0 -1

[pic 74]

A21= - -3 2 =-6+(-2)=-8 A22=+ 1 2 =-2 A23= - 1 -3 = 1[pic 75][pic 76]

-1 -2 0 -2 0 -1

A31=+ -3 2 = -16 A32= - 1 2 =4 A33=+ 1 -3 =11 [pic 77][pic 78][pic 79]

5 0 2 0 2 5

(ad)T= -10 -8 -10 Ad= -10 4 -2[pic 80][pic 81]

4 -2 4 -8 -2 1

-2 1 11 -10 4 11

A-1= 10/26 8/26 10/26 = 5/13 4/13 5/13[pic 82][pic 83]

-4/26 2/26 -4/26 -2/13 1/13 -2/13

2/26 -1/26 -11/26 1/13 1/26 -11/26

+ Actividad 8 vectores

A=-2i+3j+k. B=4i-3j+3k. C=-j+4k[pic 84][pic 85][pic 86]

- |a-b|=(-2i+3j+k) –(4i-3j+3k)=-6i+6j+4k

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