Prime matrices as defined in the following paper [Prime matrices P. F. RIVETT AND N. I. P. MACKINNON](https://www.jstor.org/stable/3616179) carry over many properties of factorization as in natural numbers to matrices over the field of naturals.


I quote the following:



> 
> A matrix in a set M of matrices is prime (naturally enough) if it is not the
> product of any other matrices in the set. We thought we would look for the
> prime matrices in the set M of all 2 x 2 matrices with entries in the non negative
> integers and with determinant 1. To our great surprise we discovered
> that: there are only two primes, and any member of M (except I) can be uniquely factorized into a product of those two.
> 
> 
> 


[EDIT (PLC): For some reason, there seems to be some confusion on which two matrices are in question. They are as follows:]


$$ P = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right),\ \ Q = 
\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $$


***My question is whether there are any other resources on prime matrices and if there has been any generalization beyond 2x2 matrices. What mathematics will I need to pursue the subject further?***


I have only three journal articles:


1. *Marenich, V. E.*, [**On prime matrices over distributive lattices**](https://doi.org/10.1007/s10958-009-9727-1), J. Math. Sci., New York 164, No. 2, 260-271 (2010); translation from Fundam. Prikl. Mat. 14, No. 7, 157–173 (2008). [Zbl 1288.15041](https://zbmath.org/1288.15041).
2. *Wang, Qianhua; Zhang, Zhongjun*, [**Algorithm for obtaining the proper relatively prime matrices of polynomial matrices**](https://doi.org/10.1080/00207178708547392), Int. J. Control 46, 769–784 (1987). [Zbl 0634.65030](https://zbmath.org/0634.65030).
3. *Rivett, P. F.; Mackinnon, N. I. P.*, [**Prime matrices**](https://doi.org/10.2307/3616179). Math. Gaz. 70, No. 454, 257–259 (1986).


Please specify whether I can get the article/journal or book for free.


Thanks.



Prime/undecomposable matrices
For a different point of view, you might like to take a look at Section 12.5 and Appendix A in the [free on-line version](http://iml.univ-mrs.fr/editions/preprint00/book/prebookdac.html) of the following book (in which you'll find some interesting open questions related to "prime matrices" of the type you described):


N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002.


