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I have a large matrix, around $10\times10$. Each individual element in the matrix itself is also a very large number, roughly of the order $10^{30}$.

I know that matrices can be used to solve linear equations. So if there is an equation $ax + by = c$ then can I denote the $10\times10$ matrix, $a$, using $x\text{ and }y$ which are just $2$ real numbers and $b\text{ and }c$ which are matrices. Can this be used to compress the larger matrix?

If not, then is there any other way that I can use so that I can use some form of short expression which can be evaluated into the larger matrix?

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    @ritratt: If there really is no pattern at all, then you won't be able to compress it. See that theorem I linked.2012-10-21

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