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$\text{Description for the following question:}$

A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By convention, $A(i,i)=0$, i.e. a person is not considered a friend of himself or herself. Let $A^k(i,j)$ refer to the $\text{(i,j)$^{th}$}$ entry in the $k^{th}$ power of the matrix A.

Suppose it is given that $A^9(i,j)>0$ for all pairs $i,j$ where $1\underline < i, j\underline <m,A^2(1,2)>0$ and $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons.

$A^2(i,i)>0$ for all $i,\;1\underline< i \underline < m.$