Matrix equations and Hilbert's tenth problem

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation Xi1 1 Xi2 2 Xik 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 number field.