A = {{4, 1, 1, 1}, 
{1, 3, -1, 
          1}, 
{1, -1, 2, 0}, 
{1, 1, 
          0, 2}}

MatrixForm[A]

Det[A[[Range[1, 3], Range[1, 3]]]]

Det[A]

Eigenvalues[A]

N[%]

L = {{2, 0, 0, 0}, 
{ .5, 1.65831, 0, 
          0}, 
{ .5, - .753778, 1.08711, 
          0}, 
{ .5,  .452267,  .0836242, 
          1.24035}}

MatrixForm[L]

U = Transpose[L]

A - L . U

MatrixForm[L . U]

CholeskyDecomposition[A]

b = {1, 2, 3, 4}

w = LinearSolve[L, b]

x = LinearSolve[U, w]

A . x

data = LUDecomposition[A]

LUBackSubstitution[data, b]

N[%]