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Let $S$ be a given matrix belonging to some matrix space over a ring. I wonder if SAGE can find matrices, say $A, B,$ in such a space, satisfying the equations $A + B = S$ and $AB = S$. Is there a way to do this? Thanks!

Let $S$ be a given matrix belonging to some matrix space over a ring. I wonder if SAGE can find matrices, say $A, B,$ in such a space, satisfying the equations $A + B = S$ and $AB = S$. Is there a way to do this? Thanks!