Untitled

Here is the previously solved system.

A = [2 0 4 3 ; -4 5 -7 -10 ; 1 15 2 -4.5 ; -2 0 2 -13]
A = 
   2.0000e+00            0   4.0000e+00   3.0000e+00
  -4.0000e+00   5.0000e+00  -7.0000e+00  -1.0000e+01
   1.0000e+00   1.5000e+01   2.0000e+00  -4.5000e+00
  -2.0000e+00            0   2.0000e+00  -1.3000e+01
b = [ 4; 9; 29; 40 ]
b = 
     4
     9
    29
    40

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

[L,U] = lufact(A);
x = backsub( U, forwardsub(L,b) )
x = 
    -3
     1
     4
    -2

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

A([2 4],:) = A([4 2],:);  
b([2 4]) = b([4 2]); 
x = A\b
x = 
  -3.0000e+00
   1.0000e+00
   4.0000e+00
  -2.0000e+00

However, LU factorization fails.

[L,U] = lufact(A);
L
L = 
   1.0000e+00            0            0            0
  -1.0000e+00   1.0000e+00            0            0
   5.0000e-01          Inf   1.0000e+00            0
  -2.0000e+00          Inf          NaN   1.0000e+00