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It is known that if A∈Mn is normal (AA*=A*A) , then AA￣=A￣A if and only if AAT=ATA. This leads to the question: do both AA￣=A￣A and AAT=ATA imply that A is normal? We give an
example to show that this is false when n=4, but we show that it is true when n=2 and n=3.