Functions computable in polynomial space

Consider nondeterministic polynomial-time Turing machine that on input x outputs a 3×3 matrix with entries from {−1,0,1} on each of its paths. Define the function f where f(x) is the upper left entry in the product of all these matrices (in an order of the paths to be made precise below). We show that the class of functions f computable as just described is exactly the class FPSPACE of integer-valued functions computable by polynomial-space Turing machines. Along the way we obtain characterizations of FPSPACE in terms of arithmetic circuits and straight-line programs.