A = {{1, 1/2, 1/3, 1/4, 1/5}, 
{1/2, 1/3, 1/4, 1/5, 
          1/6}, 
{1/3, 1/4, 1/5, 1/6, 
          1/7}, 
{1/4, 1/5, 1/6, 1/7, 
          1/8}, 
{1/5, 1/6, 1/7, 1/8, 1/9}}

MatrixForm[A]

For[i = 2,  
      i ≤ 5, i++, 
Print[
        Det[A[[Range[1, i], Range[1, i]]]]]]

N[Eigenvalues[A]]

J = Inverse[A]

MatrixForm[J]

For[i = 2,  
      i ≤ 5, i++, 
Print[
        Det[J[[Range[1, i], Range[1, i]]]]]]

N[Eigenvalues[J]]

L = {{1, 0, 0, 0, 0}, { .5,  .288675, 0, 0, 
            0}, { .333333,  .288675,  .0745356, 0, 
            0}, 
{ .25,  .259808,  .111803,  .0188982, 
            0}, 
{ .2,  .23094,  .127775,  .0377966,  .00476135}};

MatrixForm[L]

MatrixForm[L . Transpose[L]]

Linv . Transpose[Linv]

J

b = {1.234567, -10.444444, 0.000015, 
        10000.254311, -67.234888}

bnew = {1.234, -10.44, 0, 10000, -67.2}

J . b

J . bnew

U = Transpose[L];

w = LinearSolve[L, bnew]

x = LinearSolve[U, w]