Solución practica para Matrices
Solucion ejercicio 1
mi_matriz<-matrix(data = c(1,2,3,4,
5,6,7,8,
9,10,11,12),nrow = 3,byrow = TRUE)
print(mi_matriz)
## [,1] [,2] [,3] [,4]
## [1,] 1 2 3 4
## [2,] 5 6 7 8
## [3,] 9 10 11 12
mi_matriz2<-matrix(data = c(1,2,3,4,
5,6,7,8,
9,10,11,12),nrow = 3,byrow = FALSE) |> print()
## [,1] [,2] [,3] [,4]
## [1,] 1 4 7 10
## [2,] 2 5 8 11
## [3,] 3 6 9 12
Solucion ejercicio 2
ana <- c(10, 20, 30)
beto <- c(15, 25, 35)
unir_filas<- rbind(ana, beto) |> print()
## [,1] [,2] [,3]
## ana 10 20 30
## beto 15 25 35
unir_columnas<-cbind(ana, beto) |> print()
## ana beto
## [1,] 10 15
## [2,] 20 25
## [3,] 30 35
row.names(unir_filas)<-c("Maria","Jose")
unir_filas
## [,1] [,2] [,3]
## Maria 10 20 30
## Jose 15 25 35
colnames(unir_filas)<-c("examen 1","examen 2","examen 3")
unir_filas
## examen 1 examen 2 examen 3
## Maria 10 20 30
## Jose 15 25 35
Solucion ejercicio 3
# Creacion de la matriz:
set.seed(50)
(mi_matriz_aleatoria <- matrix(data = sample(x = 1:100, size = 9),
nrow = 3, byrow = TRUE)) |> print()
## [,1] [,2] [,3]
## [1,] 11 52 95
## [2,] 98 46 67
## [3,] 8 16 18
# Calcula la traspuesta:
# Sin guardar:
mi_matriz_aleatoria |> t()
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
# Con guardar:
transpuesta_mi_matriz_aleatoria<- t(mi_matriz_aleatoria) |> print()
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
#se crea y se muestra el objeto.
# Extrayendo el elemento 2,3
transpuesta_mi_matriz_aleatoria[2,3] |> print()
## [1] 16
#multiplicando la matriz por escalar:
10*transpuesta_mi_matriz_aleatoria |> print()
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
## [,1] [,2] [,3]
## [1,] 110 980 80
## [2,] 520 460 160
## [3,] 950 670 180
Solucion ejercicio 4
# Creando una matriz identidad:
matriz_identidad<-diag(x= 1, nrow = 3, ncol = 3) |> print()
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
# Creando una matriz diagonal con elementos c(5, 10, 15) en la diagonal principal.
matriz_identidad<- diag(x = c(5, 10, 15), nrow = 3) |> print()
## [,1] [,2] [,3]
## [1,] 5 0 0
## [2,] 0 10 0
## [3,] 0 0 15
Solucion ejercicio 5
# Ingreso de la matriz
m <- matrix(data = c(1,2,
3,4),nrow = 2, byrow = TRUE) |> print()
## [,1] [,2]
## [1,] 1 2
## [2,] 3 4
# Calculando la inversa
m_inversa <- solve(m) |> print()
## [,1] [,2]
## [1,] -2.0 1.0
## [2,] 1.5 -0.5
# Verificación
m %*% m_inversa |> round(digits = 0) |> print()
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1
Solucion ejercicio 6
library(matlib)
fila1<-c(2,3,5,6)
fila2<-c(0,8,1,-7)
fila3<-fila1+fila2
(matriz_para_rango<-matrix(data = c(fila1,
fila2,
fila3),nrow = 3,byrow = TRUE)) |> print()
## [,1] [,2] [,3] [,4]
## [1,] 2 3 5 6
## [2,] 0 8 1 -7
## [3,] 2 11 6 -1
rango<-matlib::R(X = matriz_para_rango) |> print()
## [1] 2
Solucion ejercicio 7
Autovalores y autovectores (Eigenvalues)
#Creando la matriz simetríca
S<-matrix(data = c(2,1,
1,2), nrow = 2, byrow = TRUE) |> print()
## [,1] [,2]
## [1,] 2 1
## [2,] 1 2
#Calcular autoValores (y tambien autoVectores)
Resultado<-eigen(S)
#autovalores
Resultado$values
## [1] 3 1
#verificar autovalores
## verificando el primer autovalor
det(S-Resultado$values[1]*diag(x = 1,2))
## [1] 0
## Verificando el segundo autovalor
det(S-Resultado$values[2]*diag(x = 1,2))
## [1] 0
Solucion ejercicio 8
library(matlib)
A<-matrix(data = c(2,3,1,
1,-2,4,
3,1,-1),nrow = 3,byrow = TRUE) |> print()
## [,1] [,2] [,3]
## [1,] 2 3 1
## [2,] 1 -2 4
## [3,] 3 1 -1
B<-matrix(data = c(1,-3,4),ncol = 1,byrow = TRUE) |> print()
## [,1]
## [1,] 1
## [2,] -3
## [3,] 4
#matriz aumentada S
S<-cbind(A,B) |> print()
## [,1] [,2] [,3] [,4]
## [1,] 2 3 1 1
## [2,] 1 -2 4 -3
## [3,] 3 1 -1 4
#Teorema de Rouche Frobenius.
matlib::R(S)==matlib::R(A)
## [1] TRUE
#Resolver el sistema:
solucion<-solve(A,B) |> print()
## [,1]
## [1,] 1
## [2,] 0
## [3,] -1
#Verificación
A%*%solucion-B
## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
Solucion ejercicio 9
library(matlib)
matlib::gaussianElimination(A,B, verbose = TRUE, fractions = TRUE)
##
## Initial matrix:
## [,1] [,2] [,3] [,4]
## [1,] 2 3 1 1
## [2,] 1 -2 4 -3
## [3,] 3 1 -1 4
##
## row: 1
##
## exchange rows 1 and 3
## [,1] [,2] [,3] [,4]
## [1,] 3 1 -1 4
## [2,] 1 -2 4 -3
## [3,] 2 3 1 1
##
## multiply row 1 by 1/3
## [,1] [,2] [,3] [,4]
## [1,] 1 1/3 -1/3 4/3
## [2,] 1 -2 4 -3
## [3,] 2 3 1 1
##
## subtract row 1 from row 2
## [,1] [,2] [,3] [,4]
## [1,] 1 1/3 -1/3 4/3
## [2,] 0 -7/3 13/3 -13/3
## [3,] 2 3 1 1
##
## multiply row 1 by 2 and subtract from row 3
## [,1] [,2] [,3] [,4]
## [1,] 1 1/3 -1/3 4/3
## [2,] 0 -7/3 13/3 -13/3
## [3,] 0 7/3 5/3 -5/3
##
## row: 2
##
## multiply row 2 by -3/7
## [,1] [,2] [,3] [,4]
## [1,] 1 1/3 -1/3 4/3
## [2,] 0 1 -13/7 13/7
## [3,] 0 7/3 5/3 -5/3
##
## multiply row 2 by 1/3 and subtract from row 1
## [,1] [,2] [,3] [,4]
## [1,] 1 0 2/7 5/7
## [2,] 0 1 -13/7 13/7
## [3,] 0 7/3 5/3 -5/3
##
## multiply row 2 by 7/3 and subtract from row 3
## [,1] [,2] [,3] [,4]
## [1,] 1 0 2/7 5/7
## [2,] 0 1 -13/7 13/7
## [3,] 0 0 6 -6
##
## row: 3
##
## multiply row 3 by 1/6
## [,1] [,2] [,3] [,4]
## [1,] 1 0 2/7 5/7
## [2,] 0 1 -13/7 13/7
## [3,] 0 0 1 -1
##
## multiply row 3 by 2/7 and subtract from row 1
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 1
## [2,] 0 1 -13/7 13/7
## [3,] 0 0 1 -1
##
## multiply row 3 by 13/7 and add to row 2
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 1
## [2,] 0 1 0 0
## [3,] 0 0 1 -1