Modelling Rules of Clan Marriage and Descendence With Matrices

In a certain society made up of four clans, A, B, C and D the rules of marriage and descent are such that a man from
clan A must marry a woman from clan B
clan B must marry a woman from clan C
clan C must marry a woman from clan D
clan D must marry a woman from clan A
Their offspring is born into their fathers clan.

Does the society allow
matrilateral cross cousin marriage?
patrilateral cross cousin marriage?
After how many generations will a man's descendant belong to his clan again?
We can represent possible marriages with the table
Husband's Clan\Wife's Clan A B C D
A 0 1 0 0
B 0 0 1 0
C 0 0 0 1
D 1 0 0 0
As a natrix this becomes  
\[M= \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array} \right)\]

We can represent the tribe taken by children with the matrix
Husband's Clan\Child's Clan A B C D
A 1 0 0 0
B 0 1 0 0
C 0 0 1 0
D 0 0 0 1
We can represent his as the matrix  
\[C= \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)\]

Because  
\[C\]
  is the identity matrix, it commutes with any other matrix, so  
\[MC=CM \neq M^TC\]
.
If the entry in the ith row and jth column of  
\[MC\]
  is not 0, then a man of clan i marries a woman of clan k and a man of clan k has children in clan j. Hence  
\[MC\]
  lists the clan of the brother in law's children.
If the entry in row i and column j of  
\[CM\]
  is not 0, is not zero, a man of clan i has children in clan k and a man of clan k marries a woman of clan j. Thus CW list the clan of the son's wife.

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