BELL, P.C. ... et al, 2008. Matrix equations and Hilbert's tenth problem. International Journal of Algebra and Computation, 18 (8), pp.1231-1241.
We show a reduction of Hilbert's tenth problem to the solvability of
the matrix equation Xi1
k = Z over non-commuting integral
matrices, where Z is the zero matrix, thus proving that the solvability
of the equation is undecidable. This is in contrast to the case whereby
the matrix semigroup is commutative in which the solvability of the
same equation was shown to be decidable in general.
The restricted problem where k = 2 for commutative matrices is
known as the \A-B-C Problem" and we show that this problem is
decidable even for a pair of non-commutative matrices over an algebraic