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# Do matrices form an abelian group under multiplication?

More precisely, I think you're asking whether the set of n X n matrices forms an abelian group under multiplication. The answer is no (assuming n>1). For example (1 0)(0 1)… = (0 1) (0 0)(0 0) (0 0), but (0 1)(1 0) = (0 0) (0 0)(0 0) (0 0)..
However, the set of n x n diagonal matrices does form an Abelian set. This is true regardless of the direction of the diagonality, right-to-left or left-to-right. Note that the resulting matrix will also be diagonal, but always right-to-left. ( Full Answer )