Clear[A,B];
A={ {0.00000001,2}, {10000000,1}};
B={2,10000001};

Clear[LU,X];
LU=LUdeco[A];
X=SolveLU[LU,B]

N[A.X - B,10]

Clear[LU,X];
LU=GaussLU[A];
X=N[GaussSol[LU,B],10]

N[A.X - B,10]

Max[(Abs[Eigenvalues[ A . Transpose[A]]])^(1/2)] * Max[(Abs[Eigenvalues[ Inverse[A].Inverse[Transpose[A]]]])^(1/2)]

Clear[H,n]; 
H[n_]:=Table[1/(i+j+1), {i,0,n},				{j,0,n}];
RHS[n_]:=Table[Sum[ 1/(i+1), {i,k,k+n}],		{k,0,n}];
X0 := Table[1, {i,0,n}];

n=10; 
StringForm["H[n]= `` RHS[n]=``", MatrixForm[H[n]],  
  MatrixForm[RHS[n]]];

Clear[LU,X];
LU=GaussLU[H[n]];   
X=GaussSol[LU,RHS[n]]

Max[Abs[H[n].X - RHS[n] ]]

Max[Abs[X-X0]]

$MachinePrecision

Clear[A,B,X,n];
  A={{0.96326,0.81321},{0.81321,0.68654}};
  B={0.88824,0.74988};
 StringForm["A= ``    B = ``", MatrixForm[A], MatrixForm[B]]

x = {0.33116,0.700000};
  Clear[i,k]; 
  n=2; M=10;
  For[k=1,k<=M,k++,
    For[i=1, i<=n, i++,
       x[[i]] = (B[[i]] - Sum[A[[i,j]] 			 x[[j]], {j,1,i-1}]
         - Sum[A[[i,j]] x[[j]],
          {j,i+1,n}])/A[[i,i]]    ];
    Print[x//N];  ];

N[GaussSeidel[A,B,10],10]

Clear[LU,X];
LU=GaussLU[A];   
X=GaussSol[LU,B]

Clear[n,Q];  
n=Length[A]; 
Q=Table[If[i >= j, A [[i,j]], 0],  								{i,n},{j,n}];  
Max[N[ Abs[Eigenvalues[
 IdentityMatrix[n]-Inverse[Q].A]], 10]]