Algebra Lineal MATRICES Y DETERMINANTES
Aicon MartinezPráctica o problema22 de Mayo de 2017
1.126 Palabras (5 Páginas)398 Visitas
INSTITUTOS TECNOLOGICOS DE MEXICO[pic 2]
INSTITUTO TECNOLOGICO DE BOCA DEL RIO[pic 3]
UNIDAD II
Materia:
Algebra Lineal
Alumno:
Martinez Domínguez Adolfo
Semestre:
2° Matutino
Carrera:
Ingeniería Civil
[pic 4]
Docente:[pic 5]
Jesus Herrera Triana
N° Control:
16990019
MATRICES Y DETERMINANTES[pic 6][pic 7]
Filas a11, a12, a13, a14, a15[pic 8][pic 9][pic 10][pic 11][pic 12][pic 13][pic 14][pic 15][pic 16]
a21, a22, a23, a24, a25[pic 17][pic 18][pic 19][pic 20][pic 21]
a31, a32, a33, a34, a35[pic 22][pic 23][pic 24][pic 25][pic 26]
a41, a42, a43, a44, a45[pic 27][pic 28][pic 29][pic 30][pic 31]
Columnas
Una matriz es un arreglo bidimensional que está compuesto por filas y columnas
[pic 32]
Cij=C Elementos Matriz de orden de 3x3
I Filas
J Columnas
Posición: = C22[pic 33]
6= C33
4=C31
12=C12
OPERACIONES DE MATRICES
“Suma, resta y multiplicación”
Suma y resta de matrices si “A”=aij y “B”= bij son matrices de orden de mxn.
Determinar la siguiente matriz A+B
Ejemplo:
[pic 34]
; B= = [pic 38][pic 35][pic 36][pic 37]
3x2 3x2
RESTA DE MATRICES
DETERMINAR A-B=?
Matriz de 3x2
B= = [pic 39][pic 40][pic 41]
Matriz de 3x2
5-(9)=5-9=4
6-(-7)=6+7=13
-1-(5)=-1-5=-6
-2-(8)=-2-8=-10
0-(12)=0-12=-12
4-(9)=4-0=4
MULTIPLICACIÓN DE MATRICES
EJEMPLO
LA MATRIZ A
B= = [pic 42][pic 43][pic 44]
3X2 2X3 2X2
[pic 45][pic 46][pic 47]
C11= (1)(3)+(2)(2)+(-3)(-1)= 3+4+3=10
C12= (1)(1)+(2)(4)+(-3)(5)= 1+8-15= -6
C21= (4)(3)+(0)(2)+(-2)(-1)= 12+0+2=14
C22= (4)(1)+(0)(4)+(-2)(5)= 4+0-10=-6
EJERCICIO 1
2X2 2X1 2X1
B= = ? = [pic 51][pic 52][pic 53][pic 54][pic 48][pic 49][pic 50]
C11= + = [pic 55][pic 56][pic 57][pic 58]
C21= + = [pic 59][pic 60][pic 61][pic 62]
INVERSA DE LA MATRIZ
A= [pic 63]
Hallar A
-3+F2=F2
-3(1) + (3)= -3+3= 0
-3(-5) + (-1)= 15-1= 14
-3(1) + (0)=-3+0=-3
-3(0) + (1)= 0+1=1
A2= [pic 64]
-5F2-14F1=F1
-5(0)-14(1)=0-14=-14
-5(14)-14(-5)=-70+70=0
-5(-3)-14(1)=15-14=1
-5(1)-14(0)=-5-0=-5
A3= [pic 65]
[pic 66][pic 67]
[pic 68][pic 69]
[pic 70][pic 71]
[pic 72][pic 73]
A4= [pic 74]
I2 A
C11 =(1) + (-5) = [pic 75][pic 76][pic 77][pic 78]
C12=[pic 79]
C13=[pic 80]
TAREA:
EJERCICIO
A==(3)(4)-(1)(2)=12-2=10[pic 81]
Hallar: [pic 82]
A1= [pic 85][pic 83][pic 84]
1F1-3F2= F2
1(3)-3(1)=3-3=0
1(2)-3(4)=2-12=-10
1(1)-3(0)=1-0=1
1(0)-3(1)=0-3=-3
A2= [pic 88][pic 86][pic 87]
10F1+2F2=F1
10(3)+2(0)=30+0=30
10(2)+2(-10)=20-20=0
10(1)+2(1)=10+2=12
10(0)+2(-3)=0-6=-6
A3= [pic 91][pic 89][pic 90]
[pic 92][pic 93]
[pic 94][pic 95][pic 96]
[pic 97][pic 98]
A4= [pic 99]
[pic 100]
= [pic 101][pic 102]
C11= (3)+(1)= [pic 103][pic 104][pic 105]
C12= (2)+(4)= [pic 106][pic 107][pic 108]
C21= (3)+(1)= [pic 109][pic 110][pic 111]
C22= (2)+(4)= [pic 112][pic 113][pic 114]
A=[pic 115]
EJERCICIO
Sumar el cuadrado de una matriz y su inversa y determinar o hallar
+ [pic 116][pic 117]
= [pic 118][pic 119]
C11=(1)(1)+(-5)(3)=1-15=-14
C12=(1)(-5)+(-5)(-1)=-5+5=0
C21=(3)(1)+(-1)(3)=3-3=0
C22=(3)(5)+(-1)(-1)=-15+1=-14
= [pic 120][pic 121][pic 122]
F2-3F1=F2
3-3(1)=3-3=0
-1-3(-5)=-1+15=14
0-3(1)=0-3=-3
1-3(0)=1-0=1
= [pic 123][pic 124][pic 125]
-14F1-5F2=F1
-14(1)-5(0)=-14-0=-14
-14(-5)-5(14)=70 -70=0
-14(1)-5(-3)=-14+15=1
-14(0)-5(1)=-0 -5=-5
= [pic 126][pic 127][pic 128]
A= [pic 129]
I2 [pic 130]
+ [pic 131][pic 132]
+ [pic 133][pic 134][pic 135]
COFACTORES
A== A’= [pic 136][pic 137][pic 138]
A== (3)(4)-(2)(1)=12-2=10[pic 140][pic 141][pic 139]
-? 2x2= orden[pic 142]
= = = [pic 143][pic 144][pic 145][pic 146][pic 147]
A== (1)(-1)-(-5)(3)= -1+15=14[pic 148]
= = [pic 149][pic 150][pic 151]
INVERSA DE LA MATRIZ
SISTEMA DE 3X3 POR METODO DE GAUS
EJERCICIO 1
[pic 152]
Celdas: 31, 21, 32, 13, 23, 12
2F3+ F1= F3
2(-1)+(2)= -2+2=0
2(0)+(1)= 0+1=1
2(1)+(-2)= 2-2=0
2(0)+(1)= 0+1=1
...