Aaux={{2,3,1,0,0,6},{3,1,0,-1,1,3}};
targetAux={0,0,0,0,-1,0};

(*Phase I*)
tab1aux=tableau[Aaux,targetAux,{3,5}]
thetas[tab1aux,1]

tab2aux=tableau[Aaux,targetAux,{3,1}]

(*
Output of Phase I is that x=(1,0,4,0) is basic feasible solution, with feasible basis {1,3}.
*)

(*Phase II*)
columnsII=Append[Range[2,5],7];
rowsII=Range[2,3];
A=tab2aux[[1]][[rowsII,columnsII]];
target={-4,-5,0,0,0};
tab1=tableau[A,target,{3,1}]
thetas[tab1,2]

tab2=tableau[A,target,{2,1}]
thetas[tab2,4]

tab3=tableau[A,target,{4,1}]

(*Optimal solution found!*)

(*Checking:*)
Maximize[4x1+5x2,2x1+3x2<=6&&3x1+x2>=3&&x1>=0&&x2>=0,{x1,x2}]