We can indicate contact between the first and second groups by a 3 x 6 matrix - 1 in the ith row and jth column indicating contact between person i in group 1 and person j in group 2, and 0 indicating no contact respectively. One such possible matrix is given.
Let
\[A= \left( \begin{array}{cccccc} 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 \end{array} \right)\]
.Similarly we can represents contact between members of the second group and members of the third group by a 6 x 7 matrix, with 1 in the ith row and jth column indicating contact bitween person i in group 2 and person j in group 3, and 0 indicating no contact respectively. One such possible matrix is given.
Let
\[B= \left( \begin{array}{ccccccc} 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 \end{array} \right)\]
.The matrix
\[AB= \left( \begin{array}{ccccccc} 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 2 & 1 & 0 & 1 & 0 \\ 2 & 0 & 1 & 1 & 0 & 2 & 0 \end{array} \right)\]
gives possible routes by which disease may be transmitted from members of group 1 to members of group 3.The 2 in the third row, first column of matrix
\[AB\]
indicates two possible routes of transmission via members of group 2.