How to figure out if any of the N lines in 3D space intersect? Let us assume that we have N lines in a three dimensional space. How can we figure out if any of these lines intersects any other line?
 
It would be great, if the problem was solved in terms of linear algebra. If it is not possible, why?

 A: According to the section about the shortest line between two lines in 3D here 
: if there are $\mu_a$ and $\mu_b$ (the formulas are given) and $P_a(\mu_a)-P_b(\mu_b)=0$, then there is an intersection point. This is how you might want to find out, if two given lines in 3D intersect. 
If you are given $N$ lines, then just make that test as often as needed, which is $$\binom{N}{2}=N(N-1)/2$$ times.
A: So far, an intuitive explanation why it is not possible was posted here and here. Namely, if we have a three dimensional real space and no equation, the solution is whole space. Each succeeding equation tends to eliminate one "free" dimension out of the remaining ones.
For example, if we want a line, which is an one dimensional object, we need to eliminate two dimensions. This requires two equations, which represent two planes in the space. However, the planes should not be parallel.  
The most algebraic answer so far comes from Gilbert Strang himself:
The case with N=2 lines makes it clear

Line 1     all points   a t + b   where a and b have 3 real components and t is a real number from -inf to +inf

Line 2      all points AT + B     where  A  and  B have 3 real components and  T is a real number ....

If those lines meet then there are numbers t and T such that   at + b = AT + B 

This is 3 eqns in 2 unknowns t and T    Create the vector x with components t and -T

   In matrix - vector form the 3 equations for x  are   [ a  A ] [ x ]     =  [B - b]

   With N lines you could check them 2 at a time -- or you could look for a better way

If the linear system has no solution, there is no intersection.
