Let's say there's a matrix of 10 billion rows filled

Are you storing these 10 billion rows of integers on disk?

That makes it much easier.

>>61923437
Open in notepad and start editing the elements

>>61923452
yes

Sort them, use run length encoding to compress and they might fit in ram. You could also summarise their values every 10K columns like redshift does.

>>61923437
Just use disk instead.
>mmap()