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I just found this problem. If you try the matrix $A^nB^m$, then your question for such matrices is equivalent to this number theory question: Can $9^n+2*9^n*2^m-12*3^n*2^m+2*3^n+2^(2*m)*9^n+2^(2*m)+2*2^m+9$ 9^n+2\cdot 9^n\cdot 2^m-12\cdot 3^n\cdot 2^m+2\cdot 3^n+4^{m}\cdot 9^n+4^{m}+2\cdot 2^m+9$be a square provided$m,n\ne 0$. Note that if we denote$3^n$by$x$,$2^m$by$y$, we get a quartic polynomial in$x,y$. I hope number theorists here can say something about itthis exponential Diophantine equation. The answer to problem with question mark is "obviously NO". To be undecidable, you should have a mass problem. For given$A,B$, you have the following problem: given a word product$W(A,B)$is it true that the matrix has an integer eigenvalue. That problem is obviously decidable. The question of whether this is true for every word$W$requires answer "yes" or "no" and is not a mass problem. You can still ask whether it is independent from ZF or even ZFC (or unprovable in the Peano arithmetic). What Bjorn had in mind is a completely different and much harder problem when you include$A, B$in the input and ask if for this$A$,$B$some product$W(A,B)$not of the form$A^n, B^m$has an integer eigenvalue. This is a mass problem which could be undecidable (although he, of course, did not prove it). But this has nothing to do with your the original question. 1 I just found this problem. If you try the matrix$A^nB^m$, then your question for such matrices is equivalent to this number theory question: Can$9^n+2*9^n*2^m-12*3^n*2^m+2*3^n+2^(2*m)*9^n+2^(2*m)+2*2^m+9$be a square provided$m,n\ne 0$. I hope number theorists here can say something about it. The answer to problem with question mark is "obviously NO". To be undecidable, you should have a mass problem. For given$A,B$, you have the following problem: given a word$W(A,B)$is it true that the matrix has an integer eigenvalue. That problem is obviously decidable. The question of whether this is true for every word$W$requires answer "yes" or "no" and is not a mass problem. You can still ask whether it is independent from ZF or even ZFC (or unprovable in the Peano arithmetic). What Bjorn had in mind is a completely different and much harder problem when you include$A, B$in the input and ask if for this$A$,$B$some product$W(A,B)$not of the form$A^n, B^m\$ has an integer eigenvalue. This is a mass problem which could be undecidable (although he, of course, did not prove it). But this has nothing to do with your original question.