Matrix Equations and Hilbert's Tenth Problem

We show a reduction of Hilbert’s tenth problem to the solvability of the matrix equation A1 1 A i2 2 · · ·A ik 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. We also show that the solvability of the matrix equation AB = C for matrices A,B,C ∈ Q is decidable in polynomial time by converting the problem to the orbit problem.