Solución Ejercicio 1
mi_matriz<-matrix(data = c(1,2,3,4,
5,6,7,8,
9,10,11,12),nrow = 3,byrow = TRUE)
mi_matriz
## [,1] [,2] [,3] [,4]
## [1,] 1 2 3 4
## [2,] 5 6 7 8
## [3,] 9 10 11 12
mi_matriz<-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
Solución Ejercicio 2
ana <- c(10,20,30)
beto <- c(15,25,35)
unir_filas <- rbind(ana,beto) |> print() #se crea el objeto unir filas y se muestra.
## [,1] [,2] [,3]
## ana 10 20 30
## beto 15 25 35
unir_columnas <- cbind(ana,beto) |> print() #se crea el objeto unir columnas y se muestra.
## ana beto
## [1,] 10 15
## [2,] 20 25
## [3,] 30 35
row.names(unir_filas)<-c("maria","jose") #se cambia el nombre de las filas del obejto unir_filas
unir_filas
## [,1] [,2] [,3]
## maria 10 20 30
## jose 15 25 35
Solución Ejercicio 3
#Creación de la matriz con números aleatorios.
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
#Calculando la transpuesta.
#Sin guardar.
mi_matriz_aleatoria |> t() #No se crea un objeto
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
#Con Guardado.
transpuesta_mi_matriz_aleatoria<-t(mi_matriz_aleatoria) |> print() #Se crea y muestra el objeto.
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
#Extrayendo el elemento de la segunda fila, tercera columna.
transpuesta_mi_matriz_aleatoria[2,3]
## [1] 16
#Multiplicando la Matriz por un 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
Solución Ejercicio 4
#Creación de Matriz de Identidad (diag())
matriz_idendidad<-diag(x = 1,nrow = 3,ncol = 3) |> print()
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
#Creación de Matriz con diagonal principal de números 5,10,15.
matriz_diagonal<-diag(x = c(5,10,15),nrow = 3) |> print()
## [,1] [,2] [,3]
## [1,] 5 0 0
## [2,] 0 10 0
## [3,] 0 0 15
Solución 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
#1- Calculando la inversa
M_inversa <- solve(M) |> print()
## [,1] [,2]
## [1,] -2.0 1.0
## [2,] 1.5 -0.5
#2- Verificación
M %*% M_inversa |>round(digits = 0) |> print()
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1
M %*% M_inversa |>round(digits = 0) |> print()
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1
#3 -
matriz_no_invertible_1 <- matrix(data = c(2,4,
0,0),nrow = 2, byrow = TRUE) |> print()
## [,1] [,2]
## [1,] 2 4
## [2,] 0 0
ifelse(det(matriz_no_invertible_1!=0),
solve(matriz_no_invertible_1),"Matriz Singular")
## [1] "Matriz Singular"
Solución 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
Solución Ejercicio 7
#Creando la Matriz simetrica
S<-matrix(data = c(2,1,
1,2),nrow = 2,byrow = TRUE) |> print()
## [,1] [,2]
## [1,] 2 1
## [2,] 1 2
#Calcular los autovalores (y también los autovectores).
resultado<-eigen(S)
#Autovalores
resultado$values
## [1] 3 1
#Verificar los autovalores.
det(S-resultado$values)
## [1] -1
#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
Solución 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
Solución Ejercicio 9
library(matlib)
matlib::gaussianElimination(A,B,verbose = TRUE,fractions = TRUE)
##
## Initial matrix:
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,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
## Warning: Function is deprecated. See latexMatrix() and Eqn() for more recent
## approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 1
## [2,] 0 1 0 0
## [3,] 0 0 1 -1