Is this a novel secret splitting method, or a different view of an existing one?
I haven't seen this one before before, but keep on reading...

It would appear to work, but you need to be a bit careful, specifically with your $mxy$ multipliers; it should be apparent that not just any random set of values would work there.

I believe that a necessary and sufficient condition would be "if you logically add a $(1, 0, 0, ..., 0)$ row to the matrix, then all submatricies consisting of $N$ rows from this extended matrix must be nonsingular" (where $N$ is the threshold number of shares); if this condition does not hold, then either there is a group of $N$ valid shares that cannot reconstruct the secret, or there is a group of $N-1$ valid shares that can.  Proving it is sufficient is straightforward; proving that it is necessary would appear to require going through a number of subcases, depending on what subset of rows was linearly dependent, and whether that subset included the artifical $(1, 0, 0, 0, ..., 0)$ row.

Blom's key distribution scheme has exactly that same requirement to its internal matrix; his key distribution scheme isn't a secret sharing scheme; instead, it's a way to distribute key shares so that any two parties can agree on a key, and no set of $N-1$ other parties can recover it.

One suggestion to make Blom's scheme work was to use the matrix:

$$
\begin{matrix}
1^0 &amp; 1^1 &amp; 1^2 &amp; 1^3\\
2^0 &amp; 2^1 &amp; 2^2 &amp; 2^3\\
3^0 &amp; 3^1 &amp; 3^2 &amp; 3^3\\
4^0 &amp; 4^1 &amp; 4^2 &amp; 4^3\\
5^0 &amp; 5^1 &amp; 5^2 &amp; 5^3\\
6^0 &amp; 6^1 &amp; 6^2 &amp; 6^3\\
\end{matrix}
$$

(where $1, 2, 3, 4, 5, 6$ are distinct nonzero field elements)

This matrix meets the criteria I mentioned, and works great for you.  However, if you look at it closely, it you use it, well, you've just rediscovered Shamir's original method :-)