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So, from a quick read of Wiki (and with a maths background), the matrix is a way of looking at how much of a product from a particular region are used by a different region.

Say there are 3 regions, A B and C. Then the matrix tells us how much of the product region A gets from regions B and C, as well as how much of its own product it uses. Similarly, it tells us what B uses from A and C, and of its own. Finally, the matrix tells us how much C gets from A and B, as well as its own. Each cell is a reference to a pair of regions (including the same region twice), with one being the exporter and the other the importer. Make sense?

Mathematically (because this is a matrix after all), the idea is that a matrix can be used to represent a set of equations. For a set of equations to be totally solvable, you need at least as many equations as you have variables. So, if you have 3 variables, you need to have at least 3 equations. This can be translated into a single matrix equation, with a square (3×3) matrix. One way to solve the equation is to use the inverse of the matrix. But you can only invert square matrices. Also, if you have more equations than variables, you either have a system that is impossible to solve or you have equations that are redundant (and don’t actually give you any new information).