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But if the matrix is not square, the left and right identities are not equivalent; they are both identity matrices, but have a different size.

How do you know that the left-identity is unique, and that the right-identity is unique?

So given an [itex] m \times n [/itex] matrix A, how do you know that the only matrix satisfying [itex] AI = A [/itex] for all A is the [itex] n \times n [/itex] identity matrix?

Is this even true? Could I possibly find multiple right-identity elements?

BiP