1. Cálculo del Rango utilizando OEF

library(reticulate)
py_install("sympy")

Using virtual environment “C:/Users/caade/Documents/.virtualenvs/r-reticulate” …

caracas::install_sympy(method = "auto")

Using virtual environment “C:/Users/caade/Documents/.virtualenvs/r-reticulate” …

library(caracas)
library(matlib)
A<-matrix(data = c("x",1,-1,
                   0,"x-2",1,
                   "2*x",1,"-x"),nrow = 3,byrow = TRUE)
Z<-A
A<-as_sym(A)
A |> texshow()

#(1/x)*F1->F1
fila<-((1 / A[1, 1]) * A[1, ]) 
A<-t(A)
A[,1]<-fila    
A<-t(A)
A |> texshow()

#-2*x*F1+F3->F3
fila<- -A[3,1]*A[1,]+A[3,]
A<-t(A)
A[,3]<-fila    
A<-t(A)
A |> texshow()

#(1/(x-2))*F2->F2
fila<-((1 / A[2, 2]) * A[2, ]) 
A<-t(A)
A[,2]<-fila    
A<-t(A)
A |> texshow()

#1*F2+F3->F3
fila<- -A[3,2]*A[2,]+A[3,]
A<-t(A)
A[,3]<-fila    
A<-t(A)
A |> texshow()

#(1/(-x+2+(1/(x-2))))*F3->F3
fila<-((1 / A[3, 3]) * A[3, ]) 
A<-t(A)
A[,3]<-fila    
A<-t(A)
A |> texshow()

Por lo tanto el rango es 3

2. Cálculo de la inversa de A

A<-as_sym(Z) #Z es copia de A :)
#Matriz aumentada
S<-cbind(A,caracas::diag_(x = 1,3))
S |> texshow()

#(1/x)*F1->F1
fila<-((1 / S[1, 1]) * S[1, ]) 
S<-t(S)
S[,1]<-fila    
S<-t(S)
S |> texshow()

#-2*x*F1+F3->F3
fila<- -S[3,1]*S[1,]+S[3,]
S<-t(S)
S[,3]<-fila    
S<-t(S)
S |> texshow()

#(1/(x-2))*F2->F2
fila<-((1 / S[2, 2]) * S[2, ]) 
S<-t(S)
S[,2]<-fila    
S<-t(S)
S |> texshow()

#(-1/x)*F2+F1->F1
fila<- -S[1,2]*S[2,]+S[1,]
S<-t(S)
S[,1]<-fila    
S<-t(S)
S |> texshow()

#1*F2+F3->F3
fila<- -S[3,2]*S[2,]+S[3,]
S<-t(S)
S[,3]<-fila    
S<-t(S)
S |> texshow()

#(1/(-x+2+(1/(x-2))))*F3->F3
fila<-((1 / S[3, 3]) * S[3, ]) 
S<-t(S)
S[,3]<-fila    
S<-t(S)
S |> texshow()

#-1/(x-2)*F2+F3->F3
fila<- -S[2,3]*S[3,]+S[2,]
S<-t(S)
S[,2]<-fila    
S<-t(S)
S |> texshow()

#-1/x-1/(x*(x-2))*F2+F3->F3
fila<- -S[1,3]*S[3,]+S[1,]
S<-t(S)
S[,1]<-fila    
S<-t(S)
S |> texshow()

Siendo la inversa de A [Simplificando las expresiones]:

simplify(S[,4:6]) |> texshow()

3. Verficando [\(A\cdot A^-1\)]

Ainv<-simplify(S[,4:6])
Ainv%*%A  |> texshow()

Ainv%*%A |> simplify() |> texshow()