A determinant of zero effectively means there are infinite solutions of x to a system of equations, therefore there is no unique solution.

This is not strictly correct.


A 1x1 matrix M = [ 0 ] has a determinant of 0, but M*x = 5 does not have a single solution x. (while a trivial example in 1D, it is easy to construct counterexamples for nD as well)

A n-dimensional matrix that is not of full rank (it's column [or row] vectors don't span a n-dimensional space) has a determinant of zero. For a system of linear equations Mx=y, where M is not of full rank, there exists an infinite number of solutions only if y is in the range of M, and the dimension of the solution space is the same as the dimension of the kernel space of M (the degree of freedom for the system).