In[29]:= 

V[n_]:=Table[Subscript[x, i]^j,{i, 0, n},{j, 0, n}];
n=3; n1=n+1; M=V[n]

In[30]:= 

LU=Simplify[LUDecomposition[M]][[1]]

In[31]:= 

L=Table[ If[i<j,LU[[j,i]], If[i==j, 1, 0]],{j,1,n1},{i,1,n1}]

In[32]:= 

U=Table[ If[i>=j,LU[[j,i]], 0],{j,1,n1},{i,1,n1}]

In[33]:= 

Simplify[Expand[L.U]]

In[36]:= 

Linv=Simplify[Inverse[L]]

In[37]:= 

Uinv=Simplify[Inverse[U]]

In[43]:= 

Simplify[{1,x,x^2,x^3}.Uinv]

In[44]:= 

Simplify[{1,x,x^2,x^3}.Uinv.Linv]

In[45]:= 

In[2]:= 

MT=Transpose[M]

In[7]:= 

LUT=Simplify[LUDecomposition[MT]][[1]]

In[8]:= 

LT=Table[ If[i<j,LUT[[j,i]], If[i==j, 1, 0]],{j,1,n1},{i,1,n1}]

In[9]:= 

UT=Table[ If[i>=j,LUT[[j,i]], 0],{j,1,n1},{i,1,n1}]

In[12]:= 

Simplify[Expand[LT.UT]]

In[13]:= 

LTinv=Simplify[Inverse[LT]]

In[14]:= 

Factor /@ Evaluate[LTinv.{1,x,x^2,x^3}]

In[15]:= 

In[17]:= 

L

In[18]:= 

LT

In[19]:= 

U

In[20]:= 

Transpose[U]

In[21]:= 

L

In[22]:= 

In[22]:= 

V[n_]:=Table[Subscript[x, i]^j,{i, 0, n},{j, 0, n}];
n=1; n1=n+1; B=V[n]

In[24]:= 

GaussJordan[B_]:=Module[{BI,m=Length[B],I},
  BI = Transpose[Catenate[{Transpose[B],IdentityMatrix[m]}]];
  For[k=1, k<m, k++,
    For[i=k+1, i<=m, i++,
      lik = -BI[[i,k]]/BI[[k,k]];
      For[j=1, j<=2m, j++,
        BI[[i,j]] = Simplify[BI[[i,j]] + lik BI[[k,j]]]
      ];
    ];
  ];
  For[k=m, k>1, k--,
    For[i=1, i<k, i++,
      lik = -BI[[i,k]]/BI[[k,k]];
      For[j=1, j<=2m, j++,
        BI[[i,j]] = Simplify[BI[[i,j]] + lik BI[[k,j]]]
      ];
    ];
  ];
  Return[BI]
];
GaussJordan[B]

In[16]:= 

I=2

In[25]:= 

V[3]

In[26]:= 

V[2]

In[28]:= 

Inverse[V[1]]

In[29]:=