Fermat's problem and Goldbach's problem over M(n)Z

The author shows that (1) for any m is an element of Z, the equation x(m) + y(m) = z(m) has a solution in SL(2)Z if and only if m is not divisible by 3 or 4; (2) for any A is an element of M(2) Z and any integer p, there are x, y is an element of M(n)Z such that x + y = A and det x = (-1)(n). det y = p; (3) for any A is an element of M(n)Z and any integer p, if n greater than or equal to 3 is odd, then there are x, y, z is an element of M(n)Z such that x + y + z = A and det x = det y = det z = p.