I presume that you know that a point, (x,y), where all three lines intersect is a solution to that system of equations and that such a system of equations is equivalent to the matrix equation , while requiring that z= 1. If that matrix has an inverse, we could multiply both sides by its inverse and get the unique solution x= 0, y= 0, z= 0.

We can only have a solution with z= 1 if the solution is not unique. That means that the matrix must not have an inverse and so its determinant must be 0.