Let a₁,..., a_m be real numbers that can be expressed as a finite product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix J_a of the vector a = (a₁,..., a_m) analogously to the Jacobian matrix J_f of a vector function f. We introduce the concept of multiplicative independence of {a₁,..., a_m} and show that J_a plays in it a similar role as J_f does in functional independence. We also present a kind of arithmetic implicit function theorem and show that J_a applies to it somewhat analogously as J_f applies to the ordinary implicit function theorem.