Solución 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
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()
## ana beto
## [1,] 10 15
## [2,] 20 25
## [3,] 30 35
rownames(unir_filas)<-c("maria","jose")
colnames(unir_filas)<-c("examen 1","examen 2","examen3")
unir_filas
## examen 1 examen 2 examen3
## maria 10 20 30
## jose 15 25 35
Solución Ejercicio 3
#Creación de la matrix
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()
## [,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 se muestra el objeto.
## [,1] [,2] [,3]
## [1,] 11 98 8
## [2,] 52 46 16
## [3,] 95 67 18
#Extrayendo el elemento 2,3
transpuesta_mi_matriz_aleatoria[2,3] |> print()
## [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:
#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 los elementos c(5, 10, 15) en la diagonal principal.
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_inversa%*%M |> 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"
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
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 tambien los autovectores
resultado<-eigen(s)
#autovalores
resultado$values
## [1] 3 1
#verificar los autovalores
det(s-resultado$values[1]*diag(x = 1, 2))==0
## [1] TRUE
#verificando el segundo autovalor
det(s-resultado$values[2]*diag(x = 1, 2))
## [1] 0
Solución ejercicio 8:
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 roche 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 #nos da un vector de ceros
## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
Solución ejercicio 9:
library(matlib)
matlib::gaussianElimination(A, B,verbose = TRUE)
##
## Initial matrix:
## Warning in printMatrix(A): 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 in printMatrix(A): 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 0.3333333
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0.3333333 -0.3333333 1.333333
## [2,] 1 -2.0000000 4.0000000 -3.000000
## [3,] 2 3.0000000 1.0000000 1.000000
##
## subtract row 1 from row 2
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0.3333333 -0.3333333 1.333333
## [2,] 0 -2.3333333 4.3333333 -4.333333
## [3,] 2 3.0000000 1.0000000 1.000000
##
## multiply row 1 by 2 and subtract from row 3
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0.3333333 -0.3333333 1.333333
## [2,] 0 -2.3333333 4.3333333 -4.333333
## [3,] 0 2.3333333 1.6666667 -1.666667
##
## row: 2
##
## multiply row 2 by -0.4285714
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0.3333333 -0.3333333 1.333333
## [2,] 0 1.0000000 -1.8571429 1.857143
## [3,] 0 2.3333333 1.6666667 -1.666667
##
## multiply row 2 by 0.3333333 and subtract from row 1
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0.000000 0.2857143 0.7142857
## [2,] 0 1.000000 -1.8571429 1.8571429
## [3,] 0 2.333333 1.6666667 -1.6666667
##
## multiply row 2 by 2.333333 and subtract from row 3
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.2857143 0.7142857
## [2,] 0 1 -1.8571429 1.8571429
## [3,] 0 0 6.0000000 -6.0000000
##
## row: 3
##
## multiply row 3 by 0.1666667
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.2857143 0.7142857
## [2,] 0 1 -1.8571429 1.8571429
## [3,] 0 0 1.0000000 -1.0000000
##
## multiply row 3 by 0.2857143 and subtract from row 1
## Warning in printMatrix(A): Function is deprecated. See latexMatrix() and Eqn()
## for more recent approaches
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.000000 1.000000
## [2,] 0 1 -1.857143 1.857143
## [3,] 0 0 1.000000 -1.000000
##
## multiply row 3 by 1.857143 and add to row 2
## Warning in printMatrix(A): 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