So it would seem we should use a 0.5x + 0.5y blur to be sure to lose information - something that makes the matrix close to singular.
Also, how to handle the boundaries? We select a box in the image and blur that; we'd want to handle the boundaries in a way that also makes sure we lose information.
>>kzrdud+(OP)
Yes, you could set each pixel in the blurred region to the same color (i.e. each matrix entry is identical) - that could be black or the average of the blurred region. "Pixelation" does the same but in smaller boxes.