0
It would form a domain, except that it fails to even be a ring. The 0 matrix has rank 0, so it is never a full rank matrix - therefore the set of full rank square matrices doesn't have an additive identity. It is true that there are no zero divisors among the full rank square matrices: if AB=0, and A has full rank, then it's invertible, so A-1AB=A-10, or B=0. Similarly, if BA=0, BAA-1=0A-1 so B=0.
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Does the ring of full rank matrices form a domain?
The full rank matrices do not include zero, so they do not form a ring in the first place.
Why do square matrices only have multiplicative inverses?
there are pseudo inverses for non-square matrices a square matrix has an inverse only if the original matrix has full rank which implies that no vector is annihilated by the matrix as a multiplicative operator
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