There is any generic mathematical expression (formula) to represent a Matrix squared ?
I search it before, but i didn't find.
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If your question is to be interpreted as asking for the image of the squaring map for square matrices of a given size n over a given field: then the only superficially obvious condition is that the determinant should be a square. Over the complex number field "almost all" square matrices are squares; but it is not true in general that all matrices are squares. That is something you can read off from the Jordan normal form.
I suppose that one can use abstract algebraic geometry to claim that the matrices that are squares are a constructible set. This gives one some idea that there is reasonable answer for a given n. But it may not be very useful.