Failure of plain LU factorization

Here is the previously solved system.

using FundamentalsNumericalComputation
A = [2 0 4 3 ; -4 5 -7 -10 ; 1 15 2 -4.5 ; -2 0 2 -13];
b = [ 4, 9, 29, 40 ];

It has a perfectly good solution, obtainable through LU factorization.

L,U = FNC.lufact(A)
x = FNC.backsub( U, FNC.forwardsub(L,b) )
4-element Array{Float64,1}:
 -3.0
  1.0
  4.0
 -2.0

If we swap the second and fourth equations, nothing essential is changed, and Julia still finds the solution.

A[[2,4],:] = A[[4,2],:]  
b[[2,4]] = b[[4,2]]
x = A\b
4-element Array{Float64,1}:
 -3.0000000000000138
  1.000000000000001
  4.000000000000004
 -1.9999999999999973

However, LU factorization fails.

L,U = FNC.lufact(A)
L
4×4 Array{Float64,2}:
  1.0   0.0    0.0  0.0
 -1.0   1.0    0.0  0.0
  0.5  Inf     1.0  0.0
 -2.0  Inf   NaN    1.0