▁the ▁alignment ▁is ▁more ▁likely ▁due ▁to ▁chance . ▁ ▁Properties ▁of ▁the ▁P AM ▁matrices ▁ ▁Sym met ry ▁of ▁the ▁P AM ▁matrices ▁ ▁While ▁the ▁mut ation ▁probability ▁matrix ▁ ▁is ▁not ▁symmetric , ▁each ▁of ▁the ▁P AM ▁matrices ▁are . ▁This ▁somewhat ▁surprising ▁property ▁is ▁a ▁result ▁of ▁the ▁relationship ▁that ▁was ▁noted ▁for ▁the ▁mut ation ▁probability ▁matrix : ▁ ▁In ▁fact , ▁this ▁relationship ▁holds ▁for ▁all ▁positive ▁integer ▁powers ▁of ▁the ▁matrix ▁: ▁ ▁{ | ▁class =" to cc ol ours ▁col laps ible ▁col lapsed " ▁width =" 6 0 %" ▁style =" text - align : left " ▁! General isation ▁of ▁property ▁to ▁positive ▁integer ▁matrix ▁powers ▁| - ▁| ▁This ▁general isation ▁can ▁be ▁proven ▁using ▁mathematical ▁induction . ▁Suppose ▁that ▁for ▁a ▁matrix ▁▁ ▁And ▁that ▁for ▁a ▁positive ▁integer ▁▁ ▁By ▁expansion ▁of ▁the ▁matrix ▁product ▁, ▁ ▁Using ▁the ▁property ▁we ▁have ▁assumed ▁of ▁the ▁matrix ▁▁ ▁And ▁using ▁the ▁property ▁for ▁the ▁matrix ▁▁ ▁In ▁this ▁case , ▁it ▁is ▁only ▁known ▁at ▁first ▁that ▁the ▁result ▁holds ▁for ▁. ▁However , ▁the ▁above ▁argument ▁shows ▁that ▁the ▁property ▁also ▁holds ▁for ▁. ▁This ▁new ▁knowledge ▁then ▁shows ▁that ▁the ▁property ▁also ▁holds ▁for ▁ ▁and ▁this ▁repe ats ▁to ▁show ▁that ▁the ▁property ▁holds ▁for ▁all ▁positive ▁integers ▁. ▁| } ▁ ▁As ▁a ▁result , ▁the ▁entries ▁of ▁the ▁P AM n ▁matrix ▁are ▁symmetric , ▁since ▁ ▁Rel ating ▁the ▁number ▁of ▁mut