But clearly there's something down here that wasn't part of the original matrix.

In this case the Yale representation contains 16 entries, compared to only 12 in the original matrix.

This indicates the potential sensitivity of the computed inverse to small changes in the original matrix.

Another variant can be obtained by extending the original matrix to negative values:

The first mitigation method is similar to a sparse sample of the original matrix, removing components that are not considered valuable.

The corresponding columns of the original matrix are a basis for the column space.

I'd like to work with the original digitized matrix.

Finally, apply a permutation W which gets back the original matrix:

We must then divide each element by the corresponding element of our original matrix.

-8 is indeed the determinant of the original matrix.