A  =  { {1, 2, 3}, 
        {4, 5, 
          6}, 
        {7, 8, 10}}

MatrixForm[A]

Y  =  Array[y, 3]

b  =  {-1, 0, 1}

Ainverse  =  Inverse[A]

MatrixForm[Ainverse]

MatrixForm[Ainverse   .  A]

Y  =  Ainverse  .  b

A . Y

MatrixForm[A . Y  -  b]

Atilde = N[A, 16]

ytilde  =  Inverse[Atilde] . N[b, 16]

MatrixForm[ytilde]

MatrixForm[Atilde . ytilde  -  b]

f[i_, j_]  =  1/(i + j)

Hilbert  =  Table[f[i, j], {i, 1, 20}, {j, 1, 20}]

MatrixForm[Hilbert]

Hinv  =  Inverse[Hilbert]

MatrixForm[Hinv]

Htilde  =  N[Hilbert, 16]

MatrixForm[Htilde]

Htildeinv  =  Inverse[Htilde]

MatrixForm[Htildeinv]

MatrixForm[Hinv - Htildeinv]

MatrixForm[A]

Eigenvalues[A]

Lambda  =  Eigenvalues[N[A, 16]]

V  =  Eigenvectors[N[A, 16]]

MatrixForm[V]

MatrixForm[Lambda]

MatrixForm[Lambda[[1]]*V[[1]]]

A . V[[1]]

MatrixForm[A . V[[1]] - Lambda[[1]]*V[[1]]]