if a be square matrix. then A+A will be a
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Assume (AT)ij=Aji(AT)ij=Aji. Then the ij−thij−thentries of A+ATA+AT are ij+jiij+ji of AA, and the ji−thji−th entries of A+ATA+AT are ji+ijji+ij of AA. By commutative prop of RR, ij+ji=ji+ijij+ji=ji+ij. So, since the ij−thij−thentries and ji−thji−th entries of A+ATA+ATare equal, A+ATA+AT is symmetric.
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