The matrix equation aXm + bYn = cI2 over M2(Z)

Let N be the set of all positive integers, and let a, b, c be nonzero integers such that gcd(a, b, c) = 1. We prove the following three results. Firstly, we show that the solvability of the matrix equation aXm + bYn = cI2, m, n ? N in M2(Z) can be reduced to the solvability of a corresponding Diophantine equation when the m atrices X and Y do not commute, i.e., XY ? YX. Alternatively, when X and Y commute, i.e., XY = YX, the solvability of this matrix equation can be reduced to the solvability of the equation axm + byn = c, m, n ? N in quadratic fields. Secondly, we determine all solutions of the matrix equationn X + Yn =cnI2, n ? N, n ? 3 in M2(Z) when X and Y do not commute. Moreover, when X and Y commute, we show that the solvability of this matrix equation can be reduced to the solvability of the equation xn + yn = cn, n ? N, n ? 3 in quadratic fields. Finally, we determine all solutions of the matrix equation aX2 + bY2 = cI2 in M2(Z).