Given two knots $K$ and $L$. With Seifert matrices $M_{K}$ and $M_{L}$ respectively, then the matrix
$\begin{bmatrix}M_K & \\ &M_L\end{bmatrix}$
is a Seifert matrix of the connected sum $K+L$.
Therefore a knot is prime if and only if it has a Seifert matrix that is not S-equivalent to a matrix of this form.
Edit: This is incorrect, every knot is S-equivalent to a prime knot.
I have two questions;
1) Is what I have said correct? No
2) My understanding is that identifying whether a certain knot is prime or not was a non-trivial question, whereas identifying S-equivalence was relatively easy. Is there some hidden difficulty I am missing?