I used to be pretty good at Matrices in school, but that was over 30 years ago and it's rusty but not forgotten. I do remember, you can reduce the rank of a matrix by decomposing it into a sum of smaller matrices multiplied by suitable row/column values, depending on how you break it down. So you might be able to convert your huge matrix into a long matrix equation of lower rank, then only load and solve as much of the equation as you can handle. This is probably the same method that roboticus talks about above.

Also as a long shot, since matrices are a set of vectors, maybe Compact and sparse bit vector might be helpful in finding a way? Maybe you can use some sort of bit model to reduce the data?

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